teatise 



earing 




lARPE 

lENCl 



Practical Treatise 



on 



Gearing 



TWELFTH EDITION 



Brown & Sharpe Mfg. Co, 
Providence, R. I., U. S. A. 

1920 



25C-11-19 



/<^-' 

/^l''-'^ i 

^ :^,^'^ 



Copyright 
1902, 1905, 1911, 1920 

BY 

Brown & Sharpe Mfg. Co. 



m 24 iS20 
g)C!.A559931 



O-^-' 



PREFACE 

This book is made for men in practical life and deals 
with questions of Gear Cutting and the construction of 
Gear Wheels in a way to meet the needs of such men 
even when they may not have the time to acquire a 
technical knowledge of the subject. 

It is also specially adapted to the use of the student 
who desires to learn methods of approximating the 
theoretical forms of gear teeth. 



CONTENTS 



Chapter I 
Pitch Circle — Pitch — Tooth — Space — Addendum or 

Face — Flank — Clearance 9 

Chapter II 
Classification — Sizing Blanks and Tooth Parts from 

Circular Pitch — Centre Distance — Pattern Gear 13 
Chapter III 

Single-Curve Gears of 30 Teeth and more 17 

Chapter IV 
Rack to Mesh with Single-Curve Gears having 30 

Teeth and more 20 

Chapter V 
Diametral Pitch — Sizing Blanks and Teeth of Spur 
Gears — Distance between the Centres of Wheels 24 
Chapter VI 
A Single-Curve Gear having 12 Teeth and an Engag- 
ing Rack showing Interference — A Gear having 
12 Teeth and an Engaging Rack without Inter- 
ference—Interchangeable Gears 29 

Chapter VII 

Double-Curve Teeth — Gear of 15 Teeth — Rack 36 

Chapter VIII 
Double-Curve Spur Gears, having More or Less 

than 15 Teeth — Annular Gears 41 

Chapter IX 

Internal Gears 45 

Chapter X 

Bevel Gear Blanks 48 

Chapter XI 
Bevel Gears — Forms and Sizes of Teeth — Cutting 

Teeth 55 

Chapter XII 

Curved Tooth or Spiral Bevel Gears 79 

Chapter XIII 
Worm Gearing — Sizing Blanks of 32 Teeth and More 83 

5 



Chapter XIV. 
Sizing Gears When the Distance Between the Centres 
and the Ratios of Speeds are Fixed — General 
Remarks — Width of Face of Spur Gears — Speed 

of Gear Cutters 101 

Chapter XV 

Spiral Gears — Calculations for Lead of Spirals 108 

Chapter XVI 
Examples in Calculations of Lead of Spirals— Angle 
of Spiral — Circumference of Spiral Gears — A Few 

Hints on Cutting 114 

Chapter XVII 
Normal Pitch of Spiral Gears — Curvature of Pitch 

Surface — Formation of Cutters 117 

Chapter XVIII 

Cutting Spiral Gears in a Universal Milling Machine . . 124 

Chapter XIX 

Spiral and Worm Gears — General Remarks 131 

Chapter XX 

Strength of Gears 135 

Chapter XXI 

Standard Proportions for Spur Gears 143 

Chapter XXII 

Tangent of Arc and Angle 145 

Chapter XXIII 
Sine and Cosine — Some of Their Applications in 

Machine Construction 151 

Chapter XXIV 
Application of Circular Functions^ Whole Diameter 
of Bevel Gear Blanks — Angles of Bevel Gear 

Blanks 157 

Chapter XXV 

Angle of Pressure 164 

Chapter XXVI 
Continued Fractions — Some Applications in Machine 

Construction 166 

Chapter XXVII 

Squares and Square Roots 172 

Tables 174 

Index 207 

6 



CHAPTER I 

Pitch Circle — Pitch — Tooth — Space — Addendum or 
Face — Flank — Clearance 



Let two cylinders, Fig. 1, touch each other, their 
axes be parallel and the cylinders be on shafts, turning 
freely. If, now, we turn one cylinder, the adhesion of 
its surface to the surface of the other cylinder will make 
that turn also. The surfaces touching each other, without 
slipping one upon the other, will evidently move through 
the same distance in a given time. This surface speed 
is called linear velocity. 



Original Cyl- 
inders. 




Fig. 1 
TANGENT CYLINDERS 

Linear Velocity is the distance a point moves along 
a line in a unit of time. 

The line described by a point in the circumference 
of either of these cylinders, as it rotates, may be called 
an arc. The length of the arc (which may be greater 
or less than the circumference of cylinder), described 
in a unit of time, is the velocity. The length, expressed 
in linear units, as inches, feet, etc., is the linear velocity. 



Linear Veloc- 



BROWN & SHARPE MFG. CO. 

The length, expressed in angular units, as degrees, is 
the angular velocity. 

If now, instead of 1° we take 360°, or one turn, as 
the angular unit, and 1 minute as the time unit, the 
loctty^'^^^'^ ^^" angular velocity will be expressed in turns or revolu- 
tions per minute. 

If these two cylinders are of the same size, one will 

make the same number of turns in a minute that the 

Relative An- othcr makes. If one cylinder is twice as large as the 

guiar Velocity, q^j^^j.^ ^j^g Smaller will make two turns while the larger 

makes one, but the linear velocity of the surface of each 

cylinder remains the same. 

This combination would be very useful in mechan- 
ism if we could be sure that one cylinder would always 
turn the other without slipping. 




C/RCLE 



Fig. 3 



Land. 



In the periphery of these two cylinders, as in Fig. 
2, cut equidistant grooves. In any grooved piece the 
places between grooves are called lands. Upon the 



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Addendum. 



lands add parts; these parts are called addenda. A 

land and its addendum is called a tooth. A toothed "^""^^ 

cylinder is called a gear. Two or more gears with teeth 

interlocking are called a train. A line c c', Fig. 2 or 3, 

between the centres of two wheels is called the line of ^vt^^^ ""^ ^^''' 

centres. A circle just touching the addenda is called 

the addendum circle. 



Gear. 
Train. 



Addendum 
Circle. 




Fig. 4 
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Pitch Circle. 



Space. 



Linear or Cir- 
cular Pitch. 



Tooth Thick- 



^ Abb re via- 
tions o f Parts 
for Teeth and 
Gears. 



To find the 
Circumference 
and Diameter 
of a Circle. 



The circumference of the cyHnders without teeth is 
called the pitch circle and exists geometrically in every 
gear. In the study of gear wheels, the problem is to so 
shape the teeth that the pitch circles will just touch each 
other without slipping. 

The groove between two teeth is called a space. In 
cut gears the width of space at pitch line and thickness 
of tooth at pitch line are equal. The distance between 
the centre of one tooth and the centre of the next tooth, 
measured along the pitch line, is the linear or circular 
pitch; that is, the linear or circular pitch is equal to a 
tooth and a space; hence, the thickness of a tooth at 
the pitch line is equal to one-half the linear or circular 
pitch. 

Let D = diameter of addendum circle. 

** D' = diameter of pitch circle. 

'' P' = linear or circular pitch. 

" if = thickness of tooth at pitch line. 

** 5 = addendum or face, also length of working 
part of tooth below pitch line or flank. 

*' 2s = T>'\ the working depth of tooth, or twice the 
addendum. 

** /= clearance or extra depth of space below work- 
ing depth. 

'' s+/= depth of space below pitch line. 

" D''+/= whole depth of space. 

** N = number of teeth in gear. 

'' 7r = 3.1416 or the circumference when diameter 
isl. 

P' is read 'T prime." D" is read ''D second." n is 
read ''pi." 

If we multiply the diameter of any circle by 7ty the 
product will be the circumference of this circle. If we 
divide the circumference of any circle by n, the quo- 
tient will be the diameter of this circle. 



12 



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CHAPTER II 

Classification — Sizing Blanks and Tooth Parts from 
Circular Pitch — Centre Distance — Pattern Gears 



If we conceive the pitch of a pair of gears to be made Elements of 
the smallest possible, we ultimately come to the con- ^^^ Teeth. 
ception of teeth that are merely lines upon the original 
pitch surfaces. These lines are called elements of the 
teeth. Gears may be classified with reference to the 
elements of their teeth, and also with reference to the 
relative position of their axes or shafts. In most gears 
the elements of teeth are either straight lines or helices 
(screw-like lines). 

This book treats upon four kinds of gears. 

First — Spur Gears; those connecting parallel shafts spur Gears. 
and whose tooth elements are straight. 

Second — Bevel Gears; those connecting shafts whose Bevei Gears, 
axes meet when sufficiently prolonged, and the elements 
of whose teeth are straight lines. In bevel gears the 
surfaces that touch each other, without slipping, are upon 
cones or parts of cones whose apexes are at the same point 
where axes of shafts meet. 

Third — Worm Gears; those connecting shafts that are worm Gears. 
not parallel and do not meet, and the elements of whose 
teeth are screw-like. 

Fourth — Spiral Gears; those connecting shafts that 
are either parallel or at an angle, but which do not 
meet, and the elements of whose teeth are helical. 

The circular pitch and number of teeth in a wheel 
being given, the diameter of the wheel and size of tooth BifS^Etc. 
parts are found as follows: 

Dividing by 3.1416 is the same as multiplying by 
3^^. Now 3.1416 = .3183; hence, multiply the circum- 
ference of a circle by .3183 and the product will be 
the diameter of the circle. Multiply the circular pitch 
by .3183 and the product will be the same part of the 

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A Diameter 
Pitch, or Mod- 
ule. 



The Module 
and the Adden- 
d u m measure 
the same, radi- 
ally. 



Diameter of 
Pitch Circle. 



Whole Diam- 
eter. 



Clearance. 



Example. 



Sizes of Blank 
and Tooth 
Parts for Gear 
of 30 teeth 13^ 
in. Circular 
Pitch. 



diameter of pitch circle that the circular pitch is of the 
circumference of pitch circle. This part is called the 
module of the pitch. There are as many m.odules con- 
tained in the diameter of a pitch circle as there are teeth 
in the wheel. 

Most mechanics make the addendum of teeth equal 
the module. Hence we can designate the module by 
the same letter as we do the addendum; that is, let s= 
the module. 

.3183 P'=s, or circular pitch multiplied by .3183=5, 
the addendum or the module. 

N5=D', or number of teeth in a wheel, multiplied 
by the module, equals diameter of pitch circle. 

(NH-2) s=D, or add 2 to the number of teeth, mul- 
tiply the sum by the addendum and the product will be the 
whole diameter. 

-Yq =/, or one-tenth of thickness of tooth at pitch line 
equals amount added to bottom of space for clearance. 

One-tenth of the thickness of tooth at pitch line is 
more than one-sixteenth of working depth, being .07854 

Example— Whed 30 teeth, 1>^'' circular pitch. P'= 
1.500''; then /=.750" or thickness of tooth equals y^". 
5=1.500" X.3183=.4775=module for l^^''?'. (See Table 
of Tooth Parts, pages 178-181.) 

D'=30x.4775"=14.325"=diameter of pitch circle. 

D = (30+2)X.4775''=15.280"=diameter of addendum 
circle, or the diameter of the blank. 

/z=^ of .7500"=.0750''=clearance at bottom of space. 

D''=2X. 4775"=. 9549''= working depth of teeth. 

D"+/=2x.4775"+.0750"=1.0299"=whole depth of 
space. 

5+/=.4775"+.0750"=.5525"=depth of space inside 
of pitch line. 

D"=25 or the working depth of teeth is equal to two 
modules. 

In making calculations it is well to retain the fourth 
place in the decimals, but when drawings are passed 



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into the workshop, three places of decimals are usually 
sufficient. 

The distance between the centres of two wheels is 
evidently equal to the radius of pitch circle of one wheel 
added to that of the other. The radius of pitch circle 
is equal to 5 multiplied by one-half the number of teeth 
in the wheel. 



Distance be- 
tween centres 
of two Gears. 




Fig. 5. SPUR GEARING 

Hence, if we know the number of teeth in two wheels, 
in mesh, and the circular pitch, to obtain the distance 
between centres we first find s; then multiply s by one- 
half the sum of number of teeth in both wheels and the 
product will be distance between centres. 

Example — What is the distance between the centres 
of two wheels 35 and 60 teeth, l}i" circular pitch? We 



15 



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first find s to be \yi" X .3183 = .3979'^ Multiplying by 
47.5 (one-half the sum of 35 and 60 teeth) we obtain 
18.9003'' as the distance between centres. 
shVinTagViS Pattern Gears should be made large enough to allow 
Gear Castings. ^^^ shrinkage in casting. In cast iron the shrinkage 
is about 1 inch in one foot. For gears one to two feet 
in diameter it is well enough to add simply -z^ of 
diameter of finished gear to the pattern. In gears 
about six inches diameter or less, the moulder will gener- 
ally rap the pattern in the sand enough to make any 
allowance for shrinkage unnecessary. In pattern gears 
the spaces between teeth should be cut wider than finished 
gear spaces to allow for rapping and to avoid having 
too much cleaning to do in order to have gears run freely. 
In cut patterns of iron it is generally enough to make 
ternatfi.^^*' spaces .015'' to .02" wider. This makes clearance .03" 
to .04" in the patterns. Some moulders might want 
.06" to .07" clearance. 

Metal patterns should be cut straight; they work 
better with no draft. It is well to leave about .005" to 
be finished from side of patterns after teeth are cut; 
this extra stock to be taken away from side where cutter 
comes through so as to take out places where stock is 
broken out. The finishing should be done with file 
or abrasive wheel, as turning in a lathe is likely to break 
out stock as badly as a cutter might do. 

If cutters are kept sharp and care is taken when coming 
through, the allowance for finishing is not necessary 
and the blanks may be finished before they are cut. 



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BROWN & SHARPE MFG. CO. 

CHAPTER III 

Single-Curve Gears of 30 Teeth and More 



Single-curve teeth are so called because they have -peet? ^^"^"'^'^ 
but one curve by theory, this curve forming both face 
and flank of tooth sides. In any gear of thirty teeth 
and more, this curve can be a single arc of a circle whose 
radius is one-fourth the radius of the pitch circle. In 
gears of thirty teeth and more, a fillet is added at bottom 
of tooth, to make it stronger, equal in radius to one-seventh 
the widest part of tooth space. 

A cutter formed to leave this fillet has the advantage 
of wearing longer than it would if brought up to a corner. 

In gears less than thirty teeth this fillet is made the 
same as just given, and sides of teeth are formed with 
more than one arc, as will be shown in Chapter VI. 

Having calculated the data of a gear of 30 teeth, Qg^r^Nfto'V^ 
yi inch circular pitch (as we did in Chapter II for Xyi" =^"- 
pitch), we proceed as follows: 

1. Draw pitch circle and point it off into parts equal 
to one-half the circular pitch. 

2. From one of these points, as at B, Fig. 6, draw 
radius to pitch circle, and upon this radius describe a 
semicircle; the diameter of this semicircle being equal 
to radius of pitch circle. Draw addendum, working 
depth and whole depth circles. 

3. From the point B, where semicircle, pitch circle 
and outer end of radius to pitch circle meet, lay off a 
distance upon semicircle equal to one-fourth the radius 
of pitch circle, shown in the figure as the chord BA. 

4. Through this new point at A, upon the semicircle, 
draw a circle concentric to pitch circle. This last is 

called the base circle, and is the one for centres of tooth Base circle 
arcs. In the system of single-curve gears we have adopted, 
the diameter of this circle is .9682" of the diameter of 

17 



Method of 
laying out. 



BROWN & SHARPE MFG. CO. 




GEAR, 30 TEETH, 
K" CIRCULAR PITCH. 
P = %" or .75" 
N = 30 
t--- .375" 
S:^ .2387" 
D"= .4775" 
5+/= .2762" 
D"+/= .5150" 
i) '1=7.1618" 
1D = 7.6392" 



Fig. 6 
SINGLE -CURVE GEAR 



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pitch circle. Thus the base circle of any gear 1 inch 
pitch diameter by this system is .9682''. If the pitch 
circle is 2" the base circle will be 1.9364''. 

5. With dividers set to one-quarter of the radius of 
pitch circle, draw arcs forming sides of teeth, placing 
one leg of the dividers in the base circle and letting 
the other leg describe an arc through a point in the 
pitch circle that was made in laying off the parts equal 
to one-half the circular pitch. Thus an arc is drawn 
about A as centre through B. 

6. With dividers set to one-seventh of the widest part 
of tooth space, draw the fillets for strengthening teeth 
at their roots. These fillet arcs should just touch the 
whole depth circle and the sides of teeth already 
described. 

Single-curve or involute gears are the only gears that invoiite^oea?- 
can run at varying distance of axes and transmit unvary- ^^s- 
ing angular velocity. This feature makes involute 
gears specially valuable for driving rolls or any rotating 
pieces, the distance of whose axes is likely to be changed. 

The assertion that gears crowd harder on bearings ^eSin^r^ °" 
when of involute than when of other forms of teeth, 
has not been proved in actual practice. 

Before taking the next chapter, the learner should make la^S'^ouueeth! 
several drawings of gears 30 teeth and more. Say 
make 35 and 70 teeth Yyi" P'. Then make 40 and 65 
teeth %" P'. 

An excellent practice will be to make drawing on 
cardboard or Bristol-board and cut teeth to lines, thus 
making paper gears; or, what is still better, make them 
of sheet metal. By placing these in mesh the learner 
can test the accuracy of his work. 



19 



BROWN & SHARPE MFG. CO. 

CHAPTER IV 

Rack to Mesh with Single-Curve Gears Having 
30 Teeth and More 



of a 'iaik. ^ "" This gear (Fig. 7) is made precisely the same as gear 
in Chapter III. 

Here the radius is drawn perpendicular to pitch line 
of rack and through one of the tooth sides, B. A semi- 
circle is drawn on each side of the radius of the pitch 
circle. 

The points A and A' are each distant from the point 
B, equal to one- fourth the radius of pitch circle and cor- 
respond to the point A in Fig. 6. 

In Fig. 7 add two lines, one passing through B and 
A and one through B and A/. These two lines form 
angles of 75>^° (degrees) with radius BO. Lines BA 
and BA' are called lines of pressure. The sides of rack 
teeth are made perpendicular to these lines. 

a Ra^ck!' '^"^ ° A Rack is a straight piece, having teeth to mesh 
with a gear. A rack may be considered as a gear of 
infinitely long radius. The circumference of a circle 
approaches a straight line as the radius increases, and 
when the radius is infinitely long any finite part of the 

of pit??Line °Jf circumference is a straight line. The pitch line of a 

^^''^' rack, then, is merely a straight line just touching the 

pitch circle of a gear meshing with the rack. The thick- 
ness of teeth, addendum and depth of teeth below pitch 
line are calculated the same as for a wheel. (For pitches 
in common use, see Tables of Tooth Parts pages 178-181.) 
The term circular pitch when applied to racks can be 
more accurately replaced by the term linear pitch. Linear 
applies strictly to a line in general while circular pertains 
to a circle. Linear pitch means the distance between the 
centres of two teeth on the pitch line whether the line 
is straight or curved. 

20 



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A rack to mesh with a single-curve gear of 30 teeth 
or more is drawn as follows: 

1. Draw straight pitch line of rack; also draw adden- dr^fng^Rack! 
dum line, working depth line and whole depth line, each 
parallel to the pitch line (see Fig. 7). 




Fig. 7 
RACK TO MESH WITH SINGLE-CURVE GEAR 
HAVING 30 TEETH AND MORE 



21 



Angle _f o r 
sides of 
Teeth. 



BROWN & SHARPE MFG. CO. 

2. Point off the pitch hne into parts equal to one- 
half the circular pitch, or=/. 

3. Through these points draw lines at an angle of 
75>^° with pitch lines, alternate lines slanting in oppo- 
site directions. The left-hand side of each rack tooth 
is perpendicular to the line BA. The right-hand side 
of each rack tooth is perpendicular to the line BA^ 

4. Add fillets at bottom of teeth equal to t of the 
width of spaces between the rack teeth at the adden- 
dum line. 

Rack The sketch, Fig. 8, will show how to approximately 
obtain angle of sides of rack teeth, directly from pitch 
line of rack, without drawing a gear in mesh with the 
rack. 




Upon the pitch line hb', draw any semicircle — ba a'b'. 
From point b lay off upon the semicircle the distance ba^ 
equal to one-quarter of the diameter of semicircle, and 
draw a straight line through b and a. 

This line, ba, makes an angle of approximately 75^"^ 
with pitch line bb', and can be one side of rack tooth. The 
same construction, b'a', will give the inclination 75>^° in 
the opposite direction for the other side of tooth. 

The sketch. Fig. 9, gives the angle of sides of a tool 
for planing out spaces between rack teeth. Upon any 
line OB draw circle OABA'. From B lay off distance 
BA and BA', each equal to one-quarter of diameter of 
the circle. 

Draw lines OA and OA'. These two lines form an 
angle of approximately 29°, and are right for inclination 
of sides of rack tool. 

22 



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Make end of rack tool .3095 of circular pitch, and then 
round the corners of the tool to leave fillets at the bottom 
of rack teeth. 

Thus, if the circular pitch of a rack is lj4" and we 

multiply by .3095, the product .4642 will be the width of 

tool at end for rack of this pitch before corners are taken 

off. This width is shown at xy. 

B 

A 



Width of Rack 
Tool at end. 




■ 
Fig. 9 

A Worm is a screw that meshes with the teeth of a gear. 

This sketch and the foregoing rule are also right for 
a worm thread tool, but a worm thread tool is not usually 
rounded for fillet. In cutting worms, leave width of 
top of thread .3354 of the circular pitch. When this is 
done, the depth of thread will be right. 



Worm Thread 
Tool. 



3354 P' 




SKETCH OF WORM THREAD 



23 



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CHAPTER V 



Diametral Pitch — Sizing Blanks and the Teeth of 
Spur Gears — Distance Between the Centres of Wheels 



^^^clJt+^ In making drawings of gears, and in cutting racks, 
cuiaT Pitch ^''^' ^^ ^^ necessary to know the circular pitch, both on account 
of spacing teeth and calculating their strength. It 
would be more convenient to express the circular pitch 
in whole inches, and the most natural divisions of an 
inch, as VT\ %'T', }4'T' and so on. But as the 

^'*f^ oJPi""! circumference of the pitch circle must contain the circular 

cumference ^ 

ciJcufa?''pitih' pitch some whole number of times, corresponding to the 

oSes''''"'^^'^ number of teeth in the gear, the diameter of the pitch 

circle will often be of a size not readily measured with a 

common rule. This is because the circumference of a 

circle is equal to 3.1416 times the diameter, or the diameter 

is equal to the circumference multiplied by .3183. 

Terms^'^'f the ^^ practlcc, it Is better that the diameter should be 

Diameter. of somc slzc couvcnicntly measured. The same applies 

to the distance between centres. Hence it is generally 

more convenient to assume the pitch in terms of the 

diameter. In Chapter II was given a definition of the 

module, and also how to obtain the module from the 

circular pitch. 

and'a%1ImS ^^ ^^^ ^^^^ assumc thc module and pass to its equiv- 

pitch. alent circular pitch. If the circumference of the pitch 

circle is divided by the number of teeth in the gear, 

the quotient will be the circular pitch. In the same 

manner, if the diameter of the pitch circle is divided by 

the number of teeth, the quotient will be the module. 

Thus, if a gear is 12 inches pitch diameter and has 48 

teeth, dividing 12" by 48, the quotient yi" \^ the module 

of this gear. In practice, the module is taken in some 

convenient part of an inch, as }4" module and so on. 

24 



BROWN & SHARPE MFG. CO. 

It is convenient in calculation to designate one of these ModSe°Diam- 

modules by s, as in Chapter II. Thus, for >^" module, ''''^^''^^■ 

s is equal to >^". Generally, in speaking of the module, 

the denominator of the fraction only is named. A 

module of \" is then called 2 diametral pitch. That is, it has 

been found more convenient to take the reciprocal of the 

module in making calculation. The reciprocal of a aN3er^ ""^ 

number is 1 divided by that number. Thus the reciprocal 

of ^ is 4, because X goes into 1 four times. 

Hence, we come to the common definition: 

Diametral Pitch is the number of teeth to one inch _ symboi f o r 

Diametral 

of diameter of pitch circle. Let this be denoted by P. Pitch. 
Thus, }i" diameter pitch we would call 4 diametral pitch 
or 4P, because there would be 4 teeth to every inch in 
the diameter of pitch circle. The circular pitch and 
the different parts of the teeth are derived from the 
diametral pitch as follows: 

%^^ = P', or 3.1416 divided by the diametral pitch Given, thepi- 

^ ./ J. ametral to find 

is equal to the circular pitch. Thus to obtain the cir- the circular 

Pitch. 

cular pitch for 4 diametral pitch, we divide 3.1416 by 4 _ , . ^. 

•^ ^ "^ To obtain Cir- 

and obtam .7854 for the circular pitch, corresponding ? u i a r pitch 

■^ r- o from D 1 a m e - 

to 4 diametral pitch. * trai pitch. 

In this case we would write P = 4, P' = .7854", s^yi". 
^" = s, or one inch divided by the number of teeth to an 
inch, gives distance on diameter of pitch circle occupied 
by one tooth or the module. The addendum or face of 
tooth is equal to the module. 

-^ = P, or one inch divided by the module equals num- 
ber of teeth to one inch or the diametral pitch. 

^ = t, or 1.5708 divided by the diametral pitch gives Given pi- 
thickness of tooth at pitch line. Thus, thickness of fjnd Thickness 

■^ ot 1 o o t n at 

teeth along the pitch line for 4 diametral pitch is .3927". Pitch Line. 

f = D', or number of teeth in a gear divided by the be?'Jf^TeShTn 

diametral pitch equals diameter of the pitch circle, ametrafpuch^t^ 

Thus for a wheel, 60 teeth, 12 P, the diameter of pitch ofiitc^'arde' 
circle will be 5 inches. 

25 



BROWN & SHARPE MFG. CO. 

Given, Num- N+2 _-^ ii<-» i -i r ^ • 

ber of Teeth in ^r" = D, or add 2 to the number or teeth m a wheel 

wheel and Dia- ii-'ii ^• ■> • i 

metrai Pitch, to and divide the sum by the diametral pitch; and the 

find Whole Di- . „, , , , , ,. . , 

ameter. Quotient Will be the whole diameter of the gear or the 

diameter of the addendum circle. Thus, for 60 teeth, 
12P, the diameter of gear blank will be SA inches. 

^, = P, or number of teeth divided by diameter of 
pitch circle in inches, gives the diametral pitch or number 
of teeth to one inch. Thus, in a wheel, 24 teeth, 3 inches 
pitch diameter, the diametral pitch is 8. 

-D~ = P, or add 2 to the number of teeth; divide the 
sum by the whole diameter of gear, and the quotient 
will be the diametral pitch. Thus, for a wheel 3i\'' 
diameter, 14 teeth, the diametral pitch is 5. 

DT = N, or diameter of pitch circle, multiplied by 
diametral pitch equals number of teeth in the gear. 
Thus, in a gear, 5 pitch, 8" pitch diameter, the num- 
ber of teeth is 40. 

DP — 2 = N or multiply the whole diameter of the 
gear by the diametral pitch, subtract 2, and the remain- 
der will be the number of teeth. 

N+2 = 5, or divide the whole diameter of a spur gear 
by the number of teeth plus two, and the quotient will 
be the addendum or module. 
The Diame- Whcu wc Say the diametral pitch we shall mean the 
number of teeth to one inch of diameter of pitch cir- 
cle, or P, (-^''=P). 

anStraj'^pu^h Whcu thc clrcular pitch is given, to find the corre- 
sponding diametral pitch, divide 3.1416 by the circular 
pitch. Thus 1.5708P is the diametral pitch correspond- 
ing to 2-inch circular pitch, C-^ = P) • 
Example. What diametral pitch corresponds to }4" circular 

pitch? Remembering that to divide by a fraction we 
multiply by the denominator and divide by the numer- 
ator, we obtain 6.2832 as the quotient of 3.1416 divided 
by }4. 6.2832P, then, is the diametral pitch correspond- 
ing to yo circular pitch. This means that in a gear of }4 
inch circular pitch there are six and two hundred and 
eighty-three one-thousandths teeth to every inch in the 

26 



from Circular 
Pitch 



BROWN & SHARPE MFG. CO. 



diameter of the pitch circle. In the table of tooth parts 
and for calculations, the diametral pitches corresponding 
to circular pitches are carried out to four places of decimals, 
but for use in the shop three places of decimals are usually 
enough. 

When two gears are in mesh, so that their pitch 
circles just touch, the distance between their axes or 
centres is equal to the sum of the radii of the two gears. 
The number of the modules between centres is equal to 
half the sum of the number of teeth in both gears. This 
principle is the same as given in Chapter II, page 13, 
but when the diametral pitch and numbers of teeth 
in two gears are given, add together the numbers of teeth in 
the two wheels and divide half the sum by the diametral 
pitch. The quotient is the centre distance. 

A gear of 20 teeth, 4P, meshes with a gear of 50 teeth; 
what is the distance between their axes or centres? Add- 
ing 50 to 20 and dividing half the sum by 4, we obtain 
^}i" as the centre distance. 

The term diametral pitch is also applied to a rack. 
Thus, a rack 3P, means a rack that will mesh with a 
gear of 3 diametral pitch. 

It will be seen that if the expression for the module 
has any number except 1 for a numerator, we cannot 
express the diametral pitch by naming the denominator 
only. Thus, if the addendum or module is A" , the 
diametral pitch will be 2yi, because 1 divided by A 
equals 2}i. 

The term module is much used where gears are made 
to metric sizes, for the reason that, the millimetre being 
so short, the module is conveniently expressed in milli- 
metres. If we know the module of a gear we can figure 
the other parts as easily as we can if we know either 
the circular pitch or the diametral pitch. The module 
is, in a sense, an actual distance, while the diametral 
pitch, or the number of teeth to an inch, is a relation or 
merely a ratio. The meaning of the module is not easily 
mistaken. 



Rule to find 
Distance be- 
tween Centres. 



Example. 



Fraction; 
Diametral 
Pitch. 



27 



BROWN & SHARPE MFG. CO. 




No. 5 AUTOMATIC GEAR CUTTING MACHINE 



Cuts spur gears to 60'' in diameter, W face. Cast iron, 
2 diametral pitch; steel, 3 diametral pitch. 

This machine is representative of our line of Spur 
Gear Cutting Machines. 

28 



BROWN & SHARPE MFG. CO. 



CHAPTER VI 



A Single-Curve Gear Having 12 Teeth and an Engaging 
Rack Showing Interference — A Gear Having 12 
Teeth and an Engaging Rack Without Inter- 
ference — Interchangeable Gears 



It has been customary to cut rack teeth with a cutter , construction 

■I 1 1 r^r- AH i ' -, ^ lor Set of gears. 

shaped to cut a 135- tooth gear. All gears having 12 teeth 
or more shaped according to the data in Chapter III, 
interchange fairly well with one another and with such 
a rack, when the pitch is not coarser than ten to the 
inch diametral (lOP), but in coarser pitches there is an 
objectionable interference as indicated in Fig. 10. 

In Fig. 10, the construction of the rack is the same as 
the construction of the rack in Chapter IV. The gear 
in Fig. 10 is drawn from the base circle out to the adden- 
dum circle, by the same method as the gear in Chapter 
III, but the spaces inside of the base circle are drawn 
as follows: 

In a gear of 12 teeth, the sides of the spaces inside of ^ f i a n k s of 

11-1 1- 1 r 1- Gears in low 

the base circle are radial for a distance, ab, equal to Numbers of 

or leeth. 

5fp or 3.5 divided by the product of the pitch by the 
number of teeth. 

With one leg of the dividers in the pitch circle in the , Method of 

drawing. 

centre of the next tooth, e, and the other leg just touching 
one of the radial lines at b, continue the tooth side into 
c, until it will touch a fillet arc, whose radius is one- 
seventh the width of the space at the addendum circle. 
The part b'c\ is an arc from the centre of the tooth g, 
etc. The flanks of the teeth or spaces in the gear. Fig. 11, 
are made the same as those in Fig. 10. 

This rule is merely conventional or not founded upon 
any principle other than the judgment of the designer, 
to produce spaces as wide as practicable, just below 

29 



BROWN & SHARPE MFG. CO. 




BROWN & SHARPE MFG. CO. 

or inside of the base circle, and then strengthen the 
flank with as large a fillet as will clear the addenda 
of any gear. If the flanks in any gear will clear the 
addenda of a rack, they will clear the addenda of any 
other gear except internal gears. An internal gear is one internal 
having teeth upon the inner side of a rim or ring. See ^^'''■• 
Chapter IX. Now, it will be seen that the gear, Fig. 10, 
has teeth too much rounded at the points or at the adden- 
dum circle. In gears of pitch coarser than 10 to the inch 
(lOP), and having fewer than 30 teeth, this rounding A?de"nTa CS 
becomes objectionable. This rounding occurs, because '^''''*^- 
in these gears arcs of circles depart too far from the true 
involute curve : — it is so much that the points of the teeth 
get no bearing on the flanks of teeth in mating wheels. 

In the gear. Fig. 11, the teeth outside the base circle 
are made as nearly true involute as a workman can get tion Sf'^x rTe 
without special machinery. This is accomplished as 
follows: draw three or four tangents to the base circle, 
ii\ jj', kk', W, letting the points of tangency on base 
circle, i', j', k', V, be about one-third or one-quarter the 
circular pitch apart; the first point /', being distant from 
/, equal to one-quarter the radius of the pitch circle. 
With the dividers set to one-quarter the radius of the 
pitch circle, placing one leg in i', draw the arc a'ij; with 
one leg in j', and radius j'j, draw jk; with one leg in k' 
and radius k'k, draw kl. Should the addendum circle 
be outside of /, the tooth side can be completed with the 
last radius, I'l. The arcs, a'ij, jk and kU together form 
a very close approximation to a true involute from the 
base circle, i'j'k'V. The exact involute for gear teeth 
is the curve made by the end of a band when unwound 
from a cylinder of the same diameter as the base circle. 

The foregoing operation of drawing the tooth sides, 
although tedious in description, is very easy of practical 
application. 

It will also be seen that the addenda of the rack teeth Rounding of 
in Fig. 10, interfere with the gear- teeth flanks, as at Rack.^" 

31 



BROWN & SHARPE MFG. CO. 




BROWN & SHARPE MFG. CO. 




BROWN & SHARPE MFG. CO. 



Templets 
necessary for 
Rounding 
Points 
Teeth. 



off 
o f 



D iagrams 



for 
Cutters. 



set of 



m, n; to avoid this interference, the teeth of the rack, 
Fig. 11, are rounded at their points or addenda. 

It is also necessary to round off the points of the invo- 
lute teeth in all gears, when they are to interchange with 
low numbered gears. In interchangeable sets of gears 
the lowest numbered pinion is usually 12. Just how 
much to round off can be learned by making templets of 
a few teeth out of thin metal or cardboard, for the gear 
and rack, or for the two gears required, and fitting the 
addenda of the teeth to clear the flanks. However 
accurate we may make a diagram, it is quite as well to 
make templets in order to shape cutters accurately. 

Fig. IIA shows a pinion whose tooth faces have been 
corrected as in the foregoing. A rack engaging with 
this pinion is also shown. 

Ordinarily, in interchangeable sets it is best to make 
cutters to corrected diagrams, as in Fig. 11 A. When 
corrected diagrams are made, as in Fig. 11 A, take the 
following: 

For 135 to rack, diagram of 135 teeth. 

" 55 " 134 teeth, " " 55 
'' 35 " 54 " " " 35 

" 26 " 34 " " " 26 

" 21 " 25 " " " 21 

.. 17 - 20 " " " 17 

- 14 " 16 " " ^L 14 

'' 12 and 13 " " " 12 

If greater accuracy is desired cutters can also be made 
for half numbers, in which case it is recommended that 
they be made as follows: 

For 80 to 134 teeth, diagram of 80 teeth. 

42 
30 
25 
19 
15 
13 



" 42 " 


54 '' 


'' 30 " 


34 '' 


'' 23 " 


25 " 


.. 19 " 


20 " 


'' 15 " 


16 " 


'' 13 





34 



BROWN & SHARPE MFG. CO. 

By making a cutter right for the lowest number of 
teeth for which it is to be used, the other teeth cut by 
this cutter will be more rounded off at the outer parts of 
the tooth faces. This rounding off is to facilitate easy 
running, the avoidance of interference and perhaps of 
noise. 




A SPUR GEAR TESTING MACHINE, WITH SPUR GEARS IN 

POSITION TO BE TESTED FOR CENTRE DISTANCE 

AND CONCENTRICITY OF THE TEETH 



35 



BROWN & SHARPE MFG. CO. 

CHAPTER VII 

Double-Curve Teeth — Gear of 15 Teeth — Rack 



Nature of III double-curve or epicycloidal teeth the formation of 
tooth sides changes at the pitch line. The outHne of the 
faces of the teeth may be traced by a point in a circle, 
rolling on the outside of the pitch circle of a gear, and the 
flanks by a point in a circle rolling on the inside of the 
pitch circle. In all gears the part of teeth outside of 
pitch line is convex; in some gears the sides of teeth 
inside pitch line are convex; in some, radial; in others, 
concave. Convex faces and concave flanks are most 
familiar to mechanics. In interchangeable sets of gears, 
one gear in each set, or of each pitch, has radial flanks. 
In the best practice, this gear has fifteen teeth. Gears with 
more than fifteen teeth, have concave flanks; gears with 
less than fifteen teeth, have convex flanks. Fifteen 
teeth is called the Base of this system. 
Construction Wc wlll first draw a gear of fifteen teeth. This fifteen- 
teeth.'' ^ "'^'' tooth construction enters into gears of any number of 
teeth and also into racks. Let the gear be 3P. Having 
obtained data, we proceed as follows: 

1. Draw pitch circle and point it off into parts equal 
to one-thirtieth of the circumference, or equal to thick- 
ness of tooth = t. 

2. From the centre, through one of these points, as 
at T, Fig. 12, draw line OTA. Draw addendum and 
whole-depth circles. 

3. About this point, T, with same radius as 15-tooth 
pitch circle, describe arcs AK and O^. For any other 
double-curve gear of 3P, the radius of arcs, AK and 
Ok, will be the same as in this 15-tooth gear = 2>^". 
In a 15-tooth gear, the arc. Ok, passes through the centre 
O, but for a gear having any other number of teeth, this 
construction arc does not pass through centre of gear. 

36 



BROWN & SHARPE MFG. CO. 



GEAR, 3 P., 15 TEETH 
P= 3 
N = 15 
P'= 1.0472" 
t= .5236" 
.3333" 
D"= .(5666" 
.<+/= .3S57" 
D"+/= .7190" 
D'= 5.0000" 
D — 5.C666" 




\ Fig. 12 

DOUBLE-CURVE GEAR 



37 



BROWN & SHARPE MFG. CO. 

Of course, the 15-tooth radius of arcs, AK and Ok, is 
always taken from the pitch we are working with. 

4. Upon these arcs on opposite sides of Hne OTA, 
lay off tooth thickness, AK and Ok, and draw line KTk. 

5. Perpendicular to KT^, draw line of pressure, 
LTP; also through O and A, draw lines AR and Or, 
perpendicular to KT^. The line of pressure is at an 
angle of 78° with the radius of gear. 

6. From O, draw a line OR to intersection of AR 
with KT^. Through point c, where OR intersects LP, 
describe a circle about the centre, O. In this circle one 
leg of dividers is placed to describe tooth faces. 

7. The radius, cd, of arc of tooth faces is the straight 
distance from c to tooth-thickness point, b, on the other 
side of radius, OT. With this radius, cb, describe both 
sides of tooth faces. 

8. Draw flanks of all teeth radial, as Oe and Of. The 
base gear, 15 teeth only, has radial flanks. 

9. With radius equal to one-seventh of the widest 
part of space, as gh, draw fillets at bottom of teeth. 

Approxima- Thc forcgolug Is a close approximation to epicycloidal 

ciddaiTeeth!'^' tccth. To gct cxact teeth, make two 15-tooth gears 

of thin metal. Made addenda long enough to come to a 

point, as at n and q. Make radial flanks, as at m and p, 

deep enough to clear addenda when gears are in mesh. 

First finish the flanks, then fit the long addenda to the 

flanks when gears are in mesh. 

standard Thcsc two tcmplct gears are exact, when the centres 

Templets. ^^^ ^^^ ^.-^j^^ distaucc apart and the teeth interlock 

without backlash. One of these templet gears can now 

be used to test any other templet gear of the same 

pitch. 

Gears and racks will be right when they run cor- 
rectly with one of these 15-tooth templet gears. Five 
or six teeth are enough to make in a gear templet. 

Double-curve Rack. Let us draw a rack 3P. Hav- 
ing obtained data of teeth we proceed as follows: 

1. Draw pitch line and point it off in parts equal 
to one-half the circular pitch. Draw addendum and 
whole depth lines. 

38 



Double-curve 
Rack. 



BROWN & SHARPE MFG. CO. 




Fig. 13 
DOUBLE-CURVE RACK 



39 



BROWN & SHARPE MFG. CO. 

2. Through one of the points, as at T, Fig. 13, draw 
Hne OTA perpendicular to pitch Hne of rack. 

3. About T make precisely the same construction as 
was made about T in Fig. 12. That is,, with radius of 
15- tooth pitch circle and centre T draw arcs Ok and 
AK; make Ok and AK equal to tooth thickness; draw 
KT^; draw Or, AR, and line of pressure, each perpendicu- 
lar to KT^. 

4. Through R and r, draw lines parallel to OA. 
Through intersections c and c' of these lines, with pressure 
line LP, draw lines parallel to pitch line. 

5. In these last lines place leg of dividers, and draw 
faces and flanks of teeth as in sketch. 

6. The radius c'd' of rack-tooth faces is the same 
length as radius cd of rack- tooth flanks, and is the straight 
distance from c to tooth-thickness point b on opposite 
side of line OA. 

7. The radius for fillet at bottom of rack teeth is 
equal to \ of the widest part of tooth space. This radius 
can be varied to suit the judgment of the designer, so long 
as a fillet does not interfere with teeth of engaging gear. 




Fig. 14 

Racks of the same pitch, to mesh with interchange- 
able gears, should be alike when placed side by side, 
and fit each other when placed together as in Fig. 14. 

In Fig. 13, a few teeth of a 15-tooth wheel are shown 
in mesh with the rack. 

40 



BROWN & SHARPE MFG. CO. 



CHAPTER VIII 



Double-Curve Spur Gears, Having More or Less 
than 15 Teeth — Annular Gears 



Let us draw two gears, 12 and 24 teeth, 4P, in mesh, ^f s^t Sf douS 
In Fig. 15 the construction hnes of the lower or 24-tooth curve Gears. 
gear are full. The upper or 12- tooth gear construction 
lines are dotted. The line of pressure, LP, and the line 
KT^ answer for both gears. The arcs AK and Ok are 
described about T. The radius of these arcs is the radius 
of pitch circle of a gear 15 teeth 4 pitch. The length 
of arcs AK and O^ is the tooth thickness for 4P. The 
line KT^ is obtained the same as in Chapter VII for 
all double-curve gears, the distances only varying as 
the pitch. Having drawn the pitch circles, the line 
KT^, and, perpendicular to KT^, the lines AR, Or and 
the line of pressure LTP, we proceed with the 24-tooth 
gear as follows: 

1. From centre C, through r, draw line intersecting 
line of pressure in m. Also draw line from centre C to 
R, crossing the line of pressure LP at c. 

2. Through m describe a circle concentric with pitch 
circle about C. This is the circle in which to place 
one leg of dividers to describe flanks of teeth. 

3. The radius, mn, of flanks is the straight distance 
from m to the first tooth-thickness point on other side 
of line of centres, CC, at v. The arc is continued to 
n, to show how constructed. This method of obtain- 
ing radius of double-curve tooth flanks applies to all 
gears having more than fifteen teeth. 

4. The construction of tooth faces is similar to 15- 
tooth wheel in Chapter VII. That is: draw a circle 
through c concentric to pitch circle; in this circle place 
one leg of dividers to draw tooth faces, the radius of 
tooth faces being cb. 

41 



BROWN & SHARPE MFG. CO. 




PINION, 12 TEETH, 
GEAR 24. TEETH, -4. P 

P=4 

N=12and24 
P'= .7854" 
t = .3927" 
S = .2500" 
D"= .5000" 
It/ = .2893" 
D"+/=.5393" 



D = 6.500 



Fig. 15 
DOUBLE-CURVE GEARS IN MESH 



42 



BROWN & SHARPE MFG. CO. 

5. The radius of fillets at roots of teeth is equal to 
one-seventh the width of space at addendum circle. 

The constructions for flanks of 12, 13 and 14 teeth are Fianksfori2. 

13 and 14 Teeth. 

similar to each other and as follows : 

1. Through centre, C, draw line from R, intersecting 
line of pressure in u. Through u draw circle about C^ In 
this circle one leg of dividers is placed for drawing flanks. 

2. The radius of flanks is the distance from u to 
the first tooth-thickness point, e, on the same side of 
CTC. This gives convex flanks. The arc is con- 
tinued to V, to show construction. 

3. This arc for flanks is continued in or toward the 
centre, only about one-sixth of the working depth (or 
is) ; the lower part of flank is similar to flanks of gear 
in Chapter VI. 

4. The faces are similar to those in 15- tooth gear, 
Chapter VII, and to the 24-tooth gear in the foregoing, 
the radius being wy] the arc is continued to x, to show 
construction. 

Annular Gears. Gears with teeth inside of a rim Annular Gears. 
or ring are called Annular or Internal Gears. The 
construction of tooth outlines is similar to the fore- 
going, but the spaces of a spur external gear become 
the teeth of an annular gear. 

It has been shown that in the system just described, 
the pinion meshing with an annular gear, must differ 
from it by at least fifteen teeth. Thus, a gear of 24 
teeth cannot work with an annular gear of 36 teeth, 
but it will work with annular gears of 39 teeth and more. 
The fillets at the roots of the teeth must be of less radius 
than in ordinary spur gears. An annular gear differing 
from its mate by less than 15 teeth can be made. This 
will be shown in Chapter IX. 

Annular gear patterns require more clearance for 
moulding than external or spur gears. 

In speaking of different sized gears, the smaller of a Pinions. 
pair is often called a pinion. 

43 



BROWN & SHARPE MFG. CO. 

The angle of pressure in all gears except involute, 
constantly changes. 78° is the pressure angle in double- 
curve, or epicycloidal gears for an instant only; in our 
example, it is 78° when one side of a tooth reaches the 
line of centres, and the pressure against teeth is applied 
in the direction of the arrows. 

The pressure angle of involute gears does not change. 
An explanation of the term angle of pressure is given 
on pages 164-165. 



44 



BROWN & SHARPE MFG. CO. 

CHAPTER IX 

Internal Gears 



Special Cut- 



in Chapter VIII, it is stated that the space of an 
internal gear is the same as the tooth of a spur gear. 
This appHes to involute or single-curve gears as well 
as to double-curve gears. 

The sides of teeth in involute internal gears are hollow- 
ing. It, however, has been customary to cut internal 
gears with spur gear cutters, a No. 1 cutter generally 
being used. This makes the teeth sides convex. Special 
cutters should be made for coarse pitch double-curve ters^Yor coarse 

. . . . Pitch. 

gears. In designmg mternal gears, it is sometimes 
necessary to depart from the system with 15-tooth base, 
so as to have the pinion differ from the wheel by less 
than 15 teeth. The rules given in Chapters VII and VIII, 
will apply in making gears on any base besides 15 teeth. 
If the base is low numbered and the pinion is small, it 
may be necessary to resort to the method given at the end 
of Chapter VII, because the teeth may be too much 
rounded at the points by following the approximate rules. 
The base must be as small as the difference between , ^ase for in- 

■'-' ternal Gear 

the internal gear and its pinion. The base can be smaller ^eeth. 
if desired. 

Let it be required to make an internal gear, and pinion 
24 and 18 teeth, 3P. Here the base cannot be more 
than 6 teeth. 

In Fig. 16 the base is 6 teeth. The arcs AK and 
Oky drawn about T, have a radius equal to the radius 
of the pitch circle of a 6-tooth gear, 3P, instead of a 
15-tooth gear, as in Chapter VIII. 

The outline of teeth of both gear and pinion is made ^.Description of 

° ^ Fig. 16. 

similar to the gear in Chapter VIII. The same letters 
refer to similar parts. The clearance circle is, however, 
drawn on the outside for the internal gear. As before 

45 



BROWN & SHARPE MFG. CO. 



V- 



GEAR, 24 TEETH. 
PINION, 18 TEETH, 3 P, 

P = 3 

N =24 and 18 
P'= 1.0472" 
t=- 5236" 
S=^ .3333' 
D"= .6666" 
S+f= .3857" 
D"+/=. .7190' 





A 


.^it 


■— — — ^^ZIL"T 


v\li ^ 


& 




\ // i 


11 


/ A /; 
/ / \ // 

./ / \ // 

// 
if 

1 1 

1 1 
1 1 

1 j 
1 j 
1 I 
1 I 
1 1 

1 1 
C 

r 



\ 



Fig, 16 
INTERNAL GEAR AND PINION IN MESH 



46 



BROWN & SHARPE MFG. CO. 

stated, the spaces of a spur wheel become the teeth of an 
internal wheel. The teeth of internal gears require but 
little for fillets at the roots; they are generally strong 
enough without fillets. The teeth of the pinion are also 
similar to the gear in Chapter VIII, substituting 6- tooth 
for 15- tooth base. To avoid confusion, it is well to make 
a complete sketch of one gear before making the other. 
The arc of action is longer in internal gears than in external 
gears. This property sometimes makes it necessary 
to give less fillets than in external gears. 

In Fig. 16 the angle KTA is 30° instead of 12°, as 
in Fig. 12. This brings the line of pressure LP at an 
angle of 60° with the radius CT, instead of 78°. A 
system of spur gears could be made upon this 6-tooth 
base. These gears would interchange, but no gear of this 
6-tooth system would mesh with a double-curve gear 
made upon the 15- tooth system in Chapter VIII. 



47 



BROWN & SHARPE MFG. CO. . 

Op, qr, and uv equal to the working depth of teeth, which 
in these gears is }4". The addendum of course is meas- 
♦ ured perpendicularly from the cone pitch lines as at kr. 

7. Draw lines Om, On, Op, Oo, Oq, Or. These lines 
give the height of teeth above the cone pitch lines as they 
approach 0, and would vanish entirely at 0. It is quite 
as well never to have the length of teeth, or face, mm' 
longer than one- third the apex distance mO, nor more 
than two and one-half times the circular pitch. 

8. Having decided upon the length of face, draw 
limiting lines m'n' perpendicular to iO, q'r' perpendicular 
to kO, and so on. 

The distance between the cone pitch lines at the 
inner ends of the teeth m'n' and q'r' is called the inner 
or smaller pitch diameter, and the circle at these points 
is called the smallest pitch circle. We now have the 
outline of a section of the gears through their axes. The 
distance mr is the whole diameter of the pinion. The 
Whole Diam- dlstaucc qo is the whole diameter of the gear. In practice 

eter of Bevel ^ o sr 

Gear Blanks ob- thcsc diamctcrs can be obtained by measuring the draw- 
tamed by Meas- . i- . . . 
uring Drawings, mg. Thc diamctcr of pmion is 3.4475" and of the gear 

6.2225". We can find the angles also by measuring the 

drawing with a protractor. In the absence of a pro-. 

tractor, templets can be cut to the drawing. The angle 

formed by line mm' with ab is the angle of face of pinion, 

in this pinion 59° 10', or 59^°. The lines qq' and gh 

give us the angle of face of gear, for this gear 22° 18', or 

22i° nearly. The angle formed by mn with ab is called 

the angle of edge of pinion, in our sketch 26° 34', or about 

26>^°. The angle of edge of gear, line qr with gh, is 

63° 26', or about 63>^°. In turning blanks to these 

angles we place one arm of the protractor or templet 

against the end of the hub, when trying angles of a blank. 

Some designers give the angles from the axes of gears, 

but it is not convenient to try blanks in this way. 

The method that we have given comes right also for 

angles as figured in compound rests. 

50 



BROWN & SHARPE MFG. CO. 




BROWN & SHARPE MFG. CO. 

When axes are at right angles, the sum of angles of edge 
in the two gears equals 90°, and the sums of angle of 
edge and face in each gear are equal. 

The angles of the axes remaining the same, all pairs 
of bevel gears of the same ratio have the same angle 
of edge; all pairs of same ratio and of same numbers 
of teeth have the same angles of both edges and faces 
independent of the pitch. Thus, in all pairs of bevel 
gears having one gear twice as large as the other, with 
axes at right angles, the angle of edge of large gear is 
63° 26', and the angle of edge of small gear is 26° 34'. 

In all pairs of bevel gears with axes at right angles, 
one gear having 24 teeth and the other gear having 12 
teeth, the angle of face of small gear is 59° 10'. 
^u'^J* ^^^u"" The following method of obtaining the whole diam- 

method of ob- ° ^ 

taming Whole etcr of bcvcl gcars is sometimes preferred : 

Diameter of 

Blanks. From k, Fig. 18, lay off; upon the cone pitch line, a 

distance kw, equal to ten times the working depth of the 
teeth = 10D". Now add iV of the shortest distance 
of w from the line ghy which is the perpendicular dotted 
line wx, to the outside pitch diameter of gear, and the 
sum will be the whole diameter of gear. In the same 
manner tV of wy, added to the outside pitch diameter of 
pinion, gives the whole diameter of pinion. The part 
added to the pitch diameter is called the diameter increment. 
Chapter XXIV gives trigonometrical methods of figur- 
ing bevel gears. In our ''Formulas in Gearing" there arc 
trigonometrical formulas for bevel gears, and also tables 
for angles and sizes. 

bSSs^^ who^se A somewhat similar construction will do for bevel 

RSht^llies.^* gears whose axes are not at right angles. 

In Fig. 19 the axes are shown at OB and OD, the angle 
BOD being less than a right angle. 

1. Parallel to OB, and at a distance from it equal 
to the radius of the gear, we draw the lines ab and cd. 

2. Parallel to OD, and at a distance from it equal 
to the radius of the pinion, we draw the lines ef and gh. 

52 



BROWN & SHARPE MFG. CO. 




J ANGLE OF AXES MORE 
THAN 90° 



Fig. 20 




^ 



INSIDE BEVEL GEAR 
AND PINION 



Fig. 21 



53 



BROWN & SHARPE MFG. CO. 

3. Now, through the point j at the intersection of 
cd and gh, we draw a Hne perpendicular to OB. This 
Hne kj, hmited by ab and cd, represents the largest 
pitch diameter of the gear. 

Through j we draw a line perpendicular to OD. This 
line jl, limited by ef and gh, represents the largest pitch 
diameter of the pinion. 

4. Through the point k at the intersection of ab 
with kj, we draw a line to 0, a line from j to 0, and 
another from /, at the intersection jl and ef to 0. These 
lines Ok, Oj, and 01, represent the cone pitch lines, as in 
Fig. 18. 

5. Perpendicular to the cone pitch lines we draw 
the lines uv, op, and qr. Upon these lines we lay off 
the addenda and working depth as in the previous figure, 
and then draw lines to the point as before. 

By a similar construction Figs. 20 and 21 can be drawn. 




STOCKING CUTTER 



54 



BROWN & SHARPE MFG. CO. 

CHAPTER XI 

Bevel Gears — Forms and Sizes of Teeth- 
Cutting Teeth 



To obtain the form of the teeth in a bevel gear we Form of bevc 

, . , , . gear teeth. 

do not lay them out upon a pitch circle, as we do m a 
spur gear, because the rolling pitch surface of a bevel 
gear, at any point, is of a longer radius of curvature 
than the actual radius of a pitch circle that passes through 
that point. Thus in Fig. 22, let fgc be the base of a 
cone about the axis OA, the diameter of the cone being 
/c, and its radius gc. Now the radius of curvature of the 
surface, at c, is evidently longer than gc, as can be seen 
in the other view at C; the full line shows the curvature 
of the surface, and the dotted lines shows the curvature 
of a circle of the radius gc. It is extremely difficult to 
represent the exact form of bevel gear teeth upon a flat 
surface, because a bevel gear is essentially spherical in 
its nature; for practical purposes we draw a line cA per- 
pendicular to Oc, letting cA reach the centre line OA, 
and take cA as the radius of a circle upon which to lay 
out the teeth. This is shown at cnm. Fig. 23. For con- 
venience the line cA is sometimes called the back cone 
radius. 

Let us take, for an example, a bevel gear and a pinion Example. 
24 and 18 teeth, 5 pitch, shafts at right angles. To 
obtain the forms of the teeth and the data for cutting, 
we need to draw a section of only a half of each gear, as 
in Fig. 23. 

1. Draw the centre lines AO and BO, then the lines 
gh and cd, and the gear blank lines as described in Chapter 
X. Extend the lines o'p' and op until they meet the 
centre lines A'B' and AB. 

2. With the radius Ac draw the arc cnm, which 
we take as the geometrical pitch circle upon which to 

55 



BROWN & SHARPE MFG. CO. 

lay out the teeth at the large end. This distance K'c' 
is taken as the radius of the geometrical pitch circle 
at the small end; to avoid confusion an arc of this circle 
is drawn at c"n'm' about A. 

3. For the pinion we have the radius Be for the geo- 
metrical pitch circle at the large end and B'c' for the 
small end: the distance B'c' is transferred to Be'''. 

4. Upon the arc cnm lay off spaces equal to the 
tooth thickness at the large pitch circle, which in our 
example is .314". Draw the outlines of the teeth as 
in previous chapters: — for single-curve teeth we draw a 
semicircle upon the radius Ac, and proceed as des- 
cribed in Chapter III. For all bevel gears that are to 
be cut with a rotary disk cutter, or a common gear cutter, 
single-curve teeth are chosen; and no attempt should 
be made to cut double-curve teeth. Double-curve 
teeth can be drawn by the directions given in Chapters 
VII and VIII. We now have the form of the teeth at 
the large end of the gear. Repeat this operation with 
the radius BC about B, and we have the form of the 
teeth at the large end of the pinion. 

5. The tooth parts at the small end are designated 
by the same letters as at the large, with the addition 
of an accent mark to each letter, as in the right-hand 
column, Fig. 23, the clearance, /, however, is usually 
the same at the small end as at the large, for conveni- 
ence in cutting the teeth. 

When cutting bevel gears with rotary cutters, the 
cutting angle is the same as the working depth angle. 
This angle is used for two reasons: — first, it is not neces- 
sary to figure the angle of the bottom; second, the 
inside of the teeth is rounded over a little more and 
this lessens the amount to be filed off at the point. When 
cut in this way, the line of the bottom of the tooth is 
parallel to the face of the mating gear and it does not 
pass through the cone apex or common point of the axes. 
tooth plrte. *^^ The sizes of the tooth parts at the small end are in 
the same proportion to those at the large end as the line 

56 



BROWN & SHARPE MFG. CO. 





• -^^ 



57 



BROWN & SHARPE MFG. CO. 

Oc' is to Oc. In our example Oc' is 2'\ and Oc is 3"; 
dividing Oc' by Oc we have i, or .666, as the ratio of 
the sizes at the small end to those at the large: f is 
.2095^' or f of .3142'', and so on. If the distance nm is 
equal to the outer tooth thickness, /, upon the arc cnm, 
the lines nA and mA will be a distance apart equal to the 
inner tooth thickness f upon the arc c''n'm'. The 
addendum, s', and the working depth, D''', are at o'c' 
and o'p'. 

6. Upon the arcs c"n'm' and c'" we draw the forms 

of the teeth of the gear and pinion at the inside. 

cJtting.^^'' ° ^ As an example of the cutting of bevel gears with rotary 

disk cutters, or common gear cutters, let us take a pair 

of 8 pitch, 12 and 24 teeth, shown in Fig. 25. 

Length of In making the drawing it is well to remember that 

tooth face. , . . ..,-,. 

nothmg IS gamed by havmg the face FE longer than 
five times the thickness of the teeth at the large pitch 
circle, and that even this is too long when it is more than 
a third of the apex distance Oc. To cut a bevel gear with 
a rotary cutter, as in Fig. 26, is at best but a compromise, 
because the teeth change pitch from end to end, so that 
the cutter, being of the right form for the large ends of the 
teeth can not be right for the small ends, and the variation 
is too great when the length of face is greater than a 
third of the apex distance Oc, Fig. 25. In the example, 
one-third of the apex distance is ^'Vbut FE is drawn 
only a half-inch, which even though rather short, has 
changed the pitch from 8 at the outside to finer than 11 
at the inside. Frequently the teeth have to be rounded 
over at the small ends by filing; the longer the teeth the 
more we have to file. If there is any doubt about the 
strength of the teeth, it is better to lengthen at the 
large end, and make the pitch coarser rather than to 
lengthen at the small end. 
Data for Thcsc data are needed before beginning to cut: 

cutting. ° 

1. The pitch and the numbers of the teeth the same 
as for spur gears. 

58 



BROWN & SHARPE MFG. CO. 




P =5. 

N =1 8 and 24 



p = 


.628" 


t' = .209" 


t = 


.314" 


S'= .133 


8 = 


.200" 


D"= .266 


D"= 


.400" 


s'+f = .165" 


8+/ = 


.231" 


D"+/ =.298' 


0"+/ = 


.431" 





Fig. 23 
BEVEL GEARS, FORM AND SIZE OF TEETH 



59 



cutters 



BROWN & SHARPE MFG. CO. 

2. The data for the cutter, as to its form: — some- 
times two cutters are needed for a pair of bevel gears. 

3. The whole depth of the tooth spaces, both at 
the outside and inside ends; D''+/ at the outside, and 
D"'-{-f at the inside. 

4. The thickness of the teeth at the outside and at 
the inside; t and t'. 

5. The height of the teeth above the pitch lines at 
the outside and inside; s and s'. 

6. The cutting angle, as applied to bevel gears cut 
with a rotary cutter is the angle that the path of the 
cutter makes with the axes of the gears. In Fig. 25 
the cutting angle for the gear cD is AOp, and that for the 
pinion is BOo. Thus the cutting angle of each gear 
equals the face angle of its mate. 

Selection of Thc form of the teeth in one of these gears differs 
so much from that in the other gear that two cutters 
are required. In determining these cutters we do not 
have to develop the forms of the gear teeth as in Fig. 
23; we need merely measure the lines Ac and Be, Fig. 25, 
and calculate the cutter forms as if these distances were 
the radii of the pitch circles of the gears to be cut. Twice 
the length Ac, in inches, multiplied by the diametral 
pitch, equals the number of teeth for which to select a 
cutter for the twenty- four- tooth gear; this number is 
about 54, which calls for a number three bevel gear 
cutter in accordance with the lists of gear cutters, page 104. 
Twice Be, multiplied by 8, equals about 13, which indicates 
a No. 8 bevel gear cutter for the pinion. This method 
of selecting cutters is based upon the idea of shaping 
the teeth as nearly right as practicable at the large end 
and then filing the small end where the cutter has not 
rounded them over enough. 

In Fig. 27 the tooth L has been cut to thickness at 
both the outer and inner pitch lines, but it must still 
be rounded at the inner end. The teeth MM have 
been filed. In thus rounding the teeth they should be 
filed above the pitch line, being careful not to file them 

60 



BROWN & SHARPE MFG. CO. 

thinner at t' as in Fig. 24 where the dotted Unes FF 
show the tooth as it is left by the cutter and the full 
lines show it after being filed to shape. 

There are several things that affect the shape of the 
teeth, so that the choice of cutters is not always so simple 
a matter as the taking of the lines Ac and Be as radii. 

In cutting a bevel gear, in the ordinary gear cutting 
machines, the finished spaces are not always of the 
same form as the cutter might be expected to make, 




Fig. 24 



because of the changes in the positions of the cutter 
and of the gear blank in order to cut the teeth of the 
right thickness at both ends. The cutter must of course 
be thin enough to pass through the small end of the spaces, 
so that the large end has to be cut to the right width 
by adjusting either the cutter or the blank sidewise, 
then rotating the blank and cutting twice around. 

Thus, in Fig. 26, a gear and a cutter are set to have 
a space widened at the large end e\ and the last chip 
to be cut off by the left side of the cutter, the cutter 



Widening the 
space at the 
large end. 



61 



BROWN & SHARPE MFG. CO. 




BEVEL GEAR DIAGRAM FOR DIMENSIONS 



62 



BROWN & SHARPE MFG. CO. 

having been moved to the right, and the blank rotated 
in the direction ot the arrow; in a universal milling 
machine the same result would be attained by moving 
the blank to the right and rotating it in the direction 
of the arrow. It may be well to remember that in setting 
to finish the side of a tooth, the tooth and the cutter 
are first separated side wise, and the blank is then rotated 
by indexing the spindle to bring the large end of the 
tooth up against the cutter. This tends not only to cut ro Jed^mofe' 
the spaces wider at the large pitch circle, but also to atroot!^ ^^^"^ 
cut off still more at the face of the tooth; that is, the 
teeth may be cut rather thin at the face and left rather 
thick at the root. This tendency is greater as a cutting 
angle BOo, Fig. 25, is smaller, or as a bevel gear approaches 
a spur gear, because when the cutting angle is small the 
blank must be rotated through a greater arc in order to 
set to cut the right thickness at the outer pitch circle. 
This can be understood by Figs. 28 and 29. Fig. 28 is a 
radial toothed clutch, which for our present purpose 
can be regarded as one extreme of a bevel gear in which the 
teeth are cut square with the axis: the dotted lines 
indicate the different positions of the cutter, the side of a 
tooth being finished by the side of the cutter that is 
on the centre line. In setting to cut these teeth there 
is the same side adjustment and rotation of the spindle 
as in a bevel gear, but there is no tendency to make a 
tooth thinner at the face than at the root. On the 
other hand, if we apply these same adjustments to a 
spur gear and cutter, Fig. 29, we shall cut the face F 
much thinner without materially changing the thick- 
ness of the root R. 

Almost all bevel gears are between the two extremes 
of Figs. 28 and 29, so that when the cutting angle BOo, 
Fig. 25, is smaller than about 30°, this change in the 
form of the spaces caused by the rotation of the blank 
may be so great as to necessitate the substitution of 
a cutter that is narrower at ee', Fig. 26, than is called 
for by the way of figuring that we have just given: thus 

63 



BROWN & SHARPE MFG. CO. 



LEFT 




RIGHT 



CUTTER MOVED IN THIS DIRECTION FOR THIS CUT 

Fig. 26 

SETTING BEVEL GEAR CUTTER OUT OF CENTRE ON BEVEL GEAR 

CUTTING MACHINE 




BROWN & SHARPE MFG. CO. 



in our own gear cutting department we might cut the 
pinion with a No. 6 cutter, instead of a No. 8. The No. 6, 
being for 17 to 20 teeth, cuts the tooth sides with a longer 
radius of curvature than the No. 8, which may necessitate 
considerable filing at the small ends of the teeth in order 
to round them over enough. Fig. 30 shows the same 

I 





Fig. 28 



Fig. 29 



gear as Fig. 27, but in this case the teeth have all been 
filed similar to MM, Fig. 27. 

Different workmen prefer different ways to com- niing the 
promise in the cutting of a bevel gear. When a blank smaii end. 
is rotated in adjusting to finish the large end of the 
teeth there need not be much filing of the small end, 




Fig. 30 
FINISHED GEAR 

65 



BROWN & SHARPE MFG. CO. 

if the cutter is right, for a pitch circle of the radius Be, Fig. 
25, which for our example is a No. 8 cutter, but the tooth 
faces may be rather thin at the large ends. This com- 
promise is preferred by nearly all workmen, because it 
does not require much filing of the teeth: — it is the same 
as is in our catalogue by which we fill any order for bevel 
cuuer°"whe°i g^ar cuttcrs, unless otherwise specified. This means 
teetji are to be ^^^^ ^^ should Send a No. 8, 8-pitch bevel gear cutter 
in reply to an order for a cutter to cut the 12- tooth 
pinion, Fig. 25; while in our own gear cutting department 
we might cut the same pinion with a No. 6, 8-pitch cutter, 
because we prefer to file the teeth at the small end after 
cutting them to the right thickness at the faces of the 
large end. We should take a No. 6 instead of a No. 8 only 
for a 12-tooth pinion that is to run with a gear two or 
three times as large. We generally step off to the next 
cutter for pinions fewer than twenty-five teeth, when 
the number for the teeth has a fraction nearly reaching 
the range of the next cutter: — thus, if twice the line Be 
in inches, Fig. 25, multiplied by the diametral pitch, 
equals 20.9, we should use a No. 5 cutter, which is for 21 
to 25 teeth inclusive. In filling an order for a gear 
cutter, we do not consider the fraction but send the 
cutter indicated by the whole number. 

Later on we will refer to other compromises that are 
made in the cutting of bevel gears. 

The sizes of the 8-pitch tooth parts, at the large end. 
Fig. 25, are copied from the table of spur gear teeth, 
pages 178-181. 
gear cutTin°g Thc dlstancc Oe' is seven- tenths of the apex distanee 
order. Q^^ g^ ^j^^^ ^j^^ gj^gs of thc tooth parts at the small end, 

except /, are seven-tenths the large. The order for 
cutting these gears goes to the workmen in this form: 

Large Gear 
P = 8 

N = 24 
D''-f/-.2696'' D'''+/=.1946'' 

^ = .1963'' f = .1374'' 

s = .1250'' 5' = .0875'' 

Cutting Angle = 59°10' = face angle of small gear. 

66 



BROWN & SHARPE MFG. CO. 



Small Gear. 
N = 12 

Cutting Angle ==22°18' = face angle of large gear. 

Fig. 34 is a front view of a gear cutting machine. 
A bevel gear blank A is held by the work spindle B. 
The cutter C is carried by the cutter slide D. The 
cutter slide carriage E can be set to the cutting angle, 
the degrees being indicated on the quadrant F. 

Fig. 36 is a plan of the machine: in this view the 
cutter slide carriage, in order to show the details a 
little plainer, is not set to an angle. 

Before beginning to cut, the cutter is set central with 
the work spindle and the dial G is set to zero, so that 
we can adjust the cutter to any required distance out 
of centre, in either direction. Set the cutter slide 
carriage E, Fig. 34, to the cutting angle of the gear, which 
for 24-teeth is 59°10'; the quadrant being divided to 
half-degrees, we estimate that 10' or I- degree more than 
59°. Mark the depth of the cut at the outside, as in Fig. 
32: — it is also well enough to mark the depth at the inside 
as a check. The thickness of the teeth at the large end 
is conveniently determined by the solid gauge, Fig. 31. 



GEAR TOOTH 
GAUGE 




■Fig. 32 



GEAR TOOTH CALIPER 

Fig. 33 



Setting 
machine. 



the 



67 



BROWN & SHARPE MFG. CO. 

The gear tooth vernier caliper, Fig. 33, will measure 
the thickness of teeth up to 2 diametral pitch. In the 
absence of the vernier caliper we can file a gauge, similar 
to Fig. 31, to the thickness of the teeth at the small end. 
side°of*t°o''oth The index having been set to divide to the right number 
being finished. ^^ ^^^ ^^^ spaccs Central with the blank, leaving a tooth 
between that is a little too thick, as in the upper part of 
Fig. 27. If the gear is of cast iron, and the pitch is not 
coarser than about 5 diametral, this is as far as we go with 
the central cuts, and we proceed to set the cutter and the 
blank to finish first one side of the teeth and then the 
other, going around only twice. The tooth has to be cut 
away more in proportion from the large than from the 
small end, which is the reason for setting the cutter out 
of centre, as in Fig. 26. 

It is important to remember that the part of the 
cutter that is finishing one side of a tooth at the pitch 
line should be central with the gear blank, in order to 
know at once in which direction to set the cutter out of 
centre. We can not readily tell how much out of centre 
to set the cutter until we have cut and tried, because 
the same part of a cutter does not cut to the pitch line 
at both ends of a tooth. As a trial distance out of centre 
we can take about one-seventh to one-sixth of the thick- 
ness of the teeth at the large end. The actual distance 
out of centre for the 12-tooth pinion is .021" : for the 
24-tooth gear, .030'', when using cutters listed in our 
catalogue. 
ceSraf'Sts/"^ After 3. Uttlc practicc a workman can set his cutter 
the trial distance out of centre, and take his first cuts, 
without any central cuts at all; but it is safer to take 
central cuts like the upper ones in Fig. 27. The depth 
of cut is partly controlled by the hand elevating shaft 
H, Fig. 36, which determines the height of the work 
spindle, and partly by the position of the cutter spindle. 
if^of Ser*"" We now set the cutter out of centre the trial distance 
by means of the cutter spindle dial shaft, I, Fig. 36. The 
trial distance can be about one-seventh the thickness 
of the tooth at the large end in a 12-tooth pinion, and 
from that to one-sixth the thickness in a 24-tooth gear 
and larger. The principle of trimming the teeth more at 

68 



out of center. 



BROWN & SHARPE MFG. CO. 




Fig. 34 

AUTOMATIC GEAR CUTTING MACHINE 

FRONT ELEVATION 



69 



Adjustments. 



BROWN & SHARPE MFG. CO. 

the large end than at the small is illustrated in Fig. 26, 
which is to move the cutter away from the tooth to be 
trimmed, and then to bring the tooth up against the 
cutter by rotating the blank in the direction of the arrow. 

The rotative adjustment of the work spindle is accom- 
plished by loosening the connection between the index 
worm and the index drive, and turning the worm: the 
connection is then fastened again. The cutter is now 
set the same distance out of centre in the other direction, 
the work spindle is adjusted to trim the other side of the 
tooth until one end is down nearly to the right thickness. 
If now the thickness of the small end is in the same 
proportidn to the large end as Oc' is to Oc, Fig. 25, we can 
at once adjust the cutter to trim the tooth to the right 
thickness. But if we find that the large end is still going 
to be too thick when the small end is right, the out of 
centre must be increased. 

It is well to remember this: too much out of centre 
leaves the small end proportionally too thick, and too 
little out of centre leaves the small end too thin. 

The amount of set-over may be calculated very closely 
from the accompanying table and formula: 

* TABLE FOR OBTAINING SET-OVER FOR CUTTING 
BEVEL GEARS 



o| 


Ratio of apex distance to width of face 


6 "§ 


1 

1 


1 


33^ 
1 


3M 
1 


4 

1 


4M 

1 


4K 
1 


4M 

1 


5 

1 


5^ 
1 


6 

1 


7 
1 


8 
1 


1 

2 
3 

4 
5 
6 

7 
8 


.254 
.266 
.266 
.275 
.280 
.311 
.289 
.275 


.254 
.268 
.268 
.280 
.285 
.318 
.298 
.286 


.255 

.271 
.271 
.285 
.290 
.323 
.308 
.296 


.256 
.272 
.273 
.287 
.293 
.328 
.316 
.309 


.257 
.273 
.275 
.291 
.295 
.330 
.324 
.319 


.257 
.274 
.278 
.293 
.296 
.334 
.329 
.331 


.257 
.274 
.280 
.296 
.298 
.337 
.334 
.338 


.258 
.275 
.282 
.298 
.300 
.340 
.338 
.344 


.258 
.277 
.283 
.298 
.302 
.343 
.343 
.352 


.259 
.279 
.286 
.302 
.307 
.348 
.350 
.361 


.260 
.280 
.287 
.305 
.309 
.352 
.360 
.368 


.262 
283 
.290 
.308 
.313 
.356 
.370 
.380 


.264 
.284 
.292 
.311 
.315 
.362 
.376 
.386 



Set-over = 



Factor from Table 



P = diametral pitch of gear to be cut. 
Tc = thickness of cutter used, measured at pitch line. 
Given as a rule, this would read: — find the factor in 
the table corresponding to the number of the cutter used and 

*From an article in Machinery by Ralph E. Flanders, prepared and edited in collabora- 
tion with Brown & Sharpe Mfg. Co. 

70 



BROWN & SHARPE MFG. CO. 



to the ratio of the apex distance to the width of face; divide 
this factor by the diametral pitch, and subtract the quotient 
from half of the thickness of the cutter at the pitch line. 




Fig. 35 

As an illustration of the use of this table in obtaining 
the set-over we will take the following example: a bevel 
gear of 24 teeth, 6 pitch, 30 degrees pitch cone angle and 
1% face. These dimensions, by the ordinary calcula- 
tions for bevel gears call for a No. 4 cutter and an apex 
distance of 4 inches. 

In order to get our factor from the table, we have to 
know the ratio of the apex distance with the length of 
face. This ratio iSi|5 = ^or about ^. The factor in 
the table for this ratio with a No. 4 cutter is 0.280. 
We next measure the cutter at the proper depth of S+/ 
for 6 pitch, which is found in the column marked "depth 
of space below pitch line" in the Table of Tooth Parts, 
pages 178-181, or by dividing 1.157 by the diametral pitch. 
This gives S+/=.1928 inch. We find by measurement 
that the thickness of the cutter at this depth is .1745 
inch. This dimension will vary with different cutters, 
and will vary in the same cutter as it is ground away, 
since formed bevel gear cutters are commonly provided 
with side relief. Substituting these values in the formula 



we have, set-over = 
required dimension. 



' — 'f = .0406 inch, which is the 



71 



mes 



BROWN & SHARPE MFG. CO. 

Mnung^Maci I^ cutting bevel gears on milling machines the work 
must be set off centre on one side of the cutter by this 
amount, taking the usual precautions to avoid errors 
from backlash. In this position the cutter is run through 
the blank, the latter being indexed for each tooth space 
until it has been cut around. (If a central or roughing 
cut has been previously taken, it will be necessary to 
line up this cut at the small end of the tooth with the 
cutter. This is done by rotating the tooth space back 
toward the cutter, either by moving the index crank 
as many holes in the dial-plate as are necessary, or by 
means of such other special provisions as may be made 
for doing this in the index head, independently of the 
dial-plate.) 

Having thus cut one side of the tooth to proper dimen- 
sions, the work must be set-over by the same amount 
the other side of the position central with the cutter, 
taking the same precautions in relation to backlash 
as before, and rotating the blank to again line up the 
cutter with the tooth space at the small end of the tooth. 
With this setting, take a trial cut. This will be found 
to leave the tooth whose side is trimmed in this operation 
a little too thick, if the cutter is thin enough, as it ought 
to be, to pass through the small end of the tooth space 
of the completed gear. This trial tooth should now be 
brought to the proper thickness by rotating the blank 
toward the cutter, moving the crank around the dial 
for the rough adjustment, and bringing it to accurate 
thickness by such means as may be provided in the head. 
In the Brown & Sharpe head, this fine adjustment is 
effected by two thumbscrews near' the hub of the index 
crank, which turn the index worm with relation to the 
crank. 

It will evidently be wise to be sure we are right before 
going ahead, as the slight approximations involved in 
the derivation of the formula may bring the setting not 
quite right, so that the thickness of the tooth at the 
large and the small ends is not what it ought to be. 

72 



BROWN & SHARPE MFG. CO. 

This point may be tested by measuring the tooth at 
both the large and the small ends with the gear tooth 
vernier caliper as shown in Fig. 33, the caliper being set 
so that the addendum at the small end is in the proper 
proportion to the addendum at the large end — (that is 
to say, that it is in the ratio ^~^ Fig. 35.) In taking 
these measurements, if the thicknesses at both the large 
and the small ends, which should be in this same ratio, are 
too great, rotate the tooth toward the cutter and take 
another cut until the proper thickness at either the 
large or small end has been obtained. If the thickness 
is right at the large end and too thick at the small end, 
the set-over is too much. I f it is right at the small end and 
too thick at the large end, the set-over is not enough, and 
should be changed accordingly, as is done by the regular 
**cut-and-try" process. The formula and table given 
herewith, however, ought to bring it near enough right 
the first time, and in the general run of work it can be 
safely relied on. 

It may be said, in this connection, that nothing but a 
true running blank, with accurate angles and diameters, 
should be used in setting up the machine. If such a 
blank cannot be found in the lot of gears to be cut, it 
will be necessary to turn one up out of wood or other 
easily worked material. Otherwise the workman is 
inviting trouble, whatever his method of setting up. 

The directions for cutting bevel gears on the milHng , ^1^}!^^ A ^ 
machine apply in modified form to the automatic gear Mlchin?.''"'"^ 
cutting machine as well. The set-over is determined in 
the same way, but instead of moving the work off centre, 
the cutter spindle is adjusted axially by means provided 
for that purpose. Some machines are provided with 
dials for reading this movement. The cutter is first 
centred as in the milling machine, and then shifted — 
first to the right, and then to the left of this central 
position. 

The rotating of the work to obtain the proper thick- 
ness of the tooth is effected by unclamping the indexing 

73 



A second 
approximation. 



BROWN & SHARPE MFG. CO. 

worm from its shaft (means usually being provided for 
this purpose) and rotating the worm until the gear 
is brought to proper position. Otherwise the operations 
are the same as for the milling machine. 

After the proper distance out of centre has been learned 
the teeth can be finish-cut by going around out of centre 
first on one side and then on the other without cutting 
any central spaces at all. The cutter spindle stops, JJ, 
can now be set to control the out of centre of the cutter, 
without having to adjust it by the dial G. If, however, a 
cast iron gear is 5-pitch or coarser it is usually well to cut 
central spaces first and then take the two out-of-centre 
cuts, going around three times in all. Steel gears should 
be cut three times around. 

Blanks are not always turned nearly enough alike to 
be cut without a different setting for different blanks. 
If the hubs vary in length the position of the cutter 
spindle has to be varied. In thus varying, the same 
depth of cut or the exact D"-\-f may not always be 
reached. A slight difference in the depth is not so 
objectionable as the incorrect tooth thickness that it 
may cause. Hence, it is well, after cutting once around 
and finishing one side of the teeth, to give careful atten- 
tion to the rotative adjustment of the work spindle 
so as to cut the right thickness. 

After a gear is cut and before the teeth are filed, it 
is not always a very satisfactory looking piece of work. 
In Fig. 27 the tooth L is as the cutter left it, and is ready 
to be filed to the shape of the teeth MM, which have 
been filed. Fig. 37 is the pair of gears that we have 
been cutting; the teeth of the 12-tooth pinion have been 
filed. 

A second approximation in cutting with a rotary 
cutter is to widen the spaces at the large end by swing- 
ing either the work spindle or the cutter slide carriage, 
so as to pass the cutter through on an angle with the 
blank sideways, called the side-angle, and not rotate 
the blank at all to widen the spaces. It is available in 

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BROWN & SHARPE MFG. CO. 

the manufacture of bevel gears in large quantities, because 
with the proper relative thickness of cutter, the tooth- 
thickness comes right by merely adjusting for the side- 
angle; but for cutting a few gears it is not much liked 
by workmen, because, in adjusting for the side-angle, 
the central setting of the cutter is usually lost, and has 
to be found by guiding into the central slot already 
cut. If the side-angle mechanism pivots about a line 
that passes very near the small end of the tooth to be 
cut, the central setting of the cutter may not be lost. 
In widening the spaces at the large end, the teeth are 
narrowed practically the same amount at the root as 
at the face, so that this side-angle method requires a 
wider cutter at ee' , Fig. 26, than the. first, or rotative 
method. The amount of filing required to correct the 
form of the teeth at the small end is about the same as in 
the first method. 

pr^imation.^^" A third approximate method consists in cutting the 
teeth right at the large end by going around at least 
twice, and then to trim the teeth at the small end and 
toward the large with another cutter, going around at 
least four times in all. This method requires skill and is 
necessarily a little slow, but it contains possibilities for 
considerable accuracy. 

proximation.^^" A fourth mcthod is to have a cutter fully as thick as 
the spaces at the small end, cut rather deeper than 
the regular depth at the large end, and go only once 
around. This is a quick method but more inaccurate 
than the three preceding: it is available in the manu- 
facture of large numbers of gears when the tooth-face 
is short compared with the apex distance. It is little 
liked, and seldom employed in cutting a few gears: it 
may require some experimenting to determine the form 
of cutter. Sometimes the teeth are not cut to the regular 
depth at the small end in order to have them thick enough, 
which may necessitate reducing the addendum of the 
teeth, 5', at the small end by turning the blank down. 
This method is extensively employed by chuck manu- 
facturers. 

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BROWN & SHARPE MFG. CO. 




Fig. 37 
FINISHED GEAR AND PINION 



77 



BROWN & SHARPE MFG. CO. 

the manufacture of bevel gears in large quantities, because 
with the proper relative thickness of cutter, the tooth- 
thickness comes right by merely adjusting for the side- 
angle; but for cutting a few gears it is not much liked 
by workmen, because, in adjusting for the side-angle, 
the central setting of the cutter is usually lost, and has 
to be found by guiding into the central slot already 
cut. If the side-angle mechanism pivots about a line 
that passes very near the small end of the tooth to be 
cut, the central setting of the cutter may not be lost. 
In widening the spaces at the large end, the teeth are 
narrowed practically the same amount at the root as 
at the face, so that this side-angle method requires a 
wider cutter at ee\ Fig. 26, than the, first, or rotative 
method. The amount of filing required to correct the 
form of the teeth at the small end is about the same as in 
the first method. 

pr^ximation.^^' A third approximate method consists in cutting the 
teeth right at the large end by going around at least 
twice, and then to trim the teeth at the small end and 
toward the large with another cutter, going around at 
least four times in all. This method requires skill and is 
necessarily a little slow, but it contains possibilities for 
considerable accuracy. 

proximatbn.^^" A fourth mcthod is to have a cutter fully as thick as 
the spaces at the small end, cut rather deeper than 
the regular depth at the large end, and go only once 
around. This is a quick method but more inaccurate 
than the three preceding: it is available in the manu- 
facture of large numbers of gears when the tooth-face 
is short compared with the apex distance. It is little 
liked, and seldom employed in cutting a few gears: it 
may require some experimenting to determine the form 
of cutter. Sometimes the teeth are not cut to the regular 
depth at the small end in order to have them thick enough, 
which may necessitate reducing the addendum of the 
teeth, s\ at the small end by turning the blank down. 
This method is extensively employed by chuck manu- 
facturers. 

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BROWN & SHARPE MFG. CO. 




Fig. Zl 
FINISHED GEAR AND PINION 



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BROWN & SHARPE MFG. CO. 



Planing of 
bevel gears. 



Mounting 
gears. 



Angles and 
sizes of bevel 
gears. 



Mitre gears. 



A machine that cuts bevel gears with a reciprocating 
motion and using a tool similar to a planer tool is called 
a gear planer and the gears so cut are said to be planed. 

One form of gear planer is that in which the prin- 
ciple embodied is theoretically correct; this machine 
originates the tooth curves without a former and is more 
often called a gear generator for this reason. Another 
form of the same class of machines is that in which the 
tool is guided by a former. 

The gear generator is more often used on the smaller 
gears while the planer type, which uses a former for 
getting the shape of teeth, is used on the larger pitch 
gears. 

If gears are not correctly mounted in the place where 
they are to run, they might as well not be planed. In 
fact, after taking pains in the cutting of any gear, when 
we come to the mounting of it we should keep right on 
taking pains. i^ ^ *|^ ^ ^ 

The method of obtaining the sizes and angles pertaining 
to bevel gears by measuring a drawing is quite convenient, 
and with care is fairly accurate. Its accuracy depends, 
of course, upon the careful measuring of a good drawing. 
We may say, in general, that in measuring a diagram, while 
we can hardly obtain data mathematically exact, we are 
not likely to make wild mistakes. We, however, calcu- 
late the data without any measuring of a drawing. In 
the ''Formulas in Gearing" there are also tables pertain- 
ing to bevel gears. 

When each gear of a pair of bevel gears is of the same 
size and the gears connect shafts that are at right angles, 
the gears are called ''mitre gears'' and one cutter will 
answer for both. 



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BROWN Si SHARPE MFG. CO. 

CHAPTER XII 

Curved Tooth or Spiral Bevel Gears 



These gears have been developed for use in automobile 
rear axles and are used to a certain extent for other 
purposes, requiring an especially smooth running drive. 

In these gears the axes of the pinion and the gear inter- 
sect as in regular bevel gears having radial teeth. 

The object in cutting curved teeth is to obtain a 
smoother and quieter drive and to increase the number 
of teeth in contact at a given instant. 

The bearing between the teeth, at any instant, instead 
of being along straight lines, as in bevel gears having radial 
teeth, runs from the base of the tooth at one end toward 
the top at the other end in a diagonal line as in herring- 
bone gears. This produces uniform wear and helps 
preserve the original tooth outline. 

Another advantage sometimes claimed is that the 
position of the pinion can be adjusted a greater amount 
than is possible with regular bevel gears without seriously 
affecting their running qualities. ^ 

The teeth of the gear are inclined, or curved, in the 
opposite direction to those of the pinion; one being right- 
hand and the other left. This inclination of the teeth 
causes the pinion to thrust in or out, according to the 
hand of the teeth and the direction of rotation, while 
the pitch angle causes the pinion to thrust out from its 
apex. Thus, when these forces act in opposite directions, 
the load on the thrust bearings is reduced and is equal 
to their difference, but when the direction of rotation of 
the gears is reversed, the thrust load on the bearing is 
increased and is equal to the sum of the above mentioned 
forces. 

No change is necessary in the general design of the 
gears when changing from the regular type to those 

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BROW^N & SHARPE MFG. CO. 




Fig. 38 
CURVED TOOTH BEVEL GEARS 



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BROWN & SHARPE MFG. CO. 

having curved teeth. The same blanks can be used 
with the same number of teeth. The only difference in 
the tooth measurements is that the normal thickness 
is, of course, reduced. 

Fig. 38 shows a pair of curved tooth bevel gears. 
These are cut on a machine especially designed for this 
purpose and the cutting is done by means of an inserted 
tooth cutter the blades of which cut upon its face and thus 
give the tooth its curved shape. 



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BROWN & SHARPE MFG. CO. 




Fig. 39 
WORM GEARING 



NUMBER OF TEETH, 54. 
THROAT DIAMETER, 44. 59' 



CIRCULAR PITCH, tVi" . 
OUTSIDE DIAMETER, 46' 



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BROWN & SHARPE MFG. CO. 



CHAPTER XIII 

Worm Gearing — Sizing Blanks of 32 Teeth and More 



A worm is a screw made to mesh with the teeth of ^'''"°^- 
a wheel called a worm wheel, Fig. 39. As implied at the 
end of Chapter IV, a section of a worm through its axis is, 
in outline, the same as a rack of corresponding pitch. 
This outline can be made either to mesh with single or 
double-curve gear teeth; but worms are usually made 
for single-curve, because, the sides of involute rack- 
teeth being straight (see Chapter IV), the tool for cutting 
a worm thread is more easily made. The thread tool 
is not usually rounded for giving fillets at bottom of 
worm thread. 

The axis of a worm is usually at right angles to the 
axis of a worm wheel : — no other angle of axis is treated 
of in this book. 

The rules for circular pitch apply in the size of tooth 
parts and diameter of pitch circle of worm wheel. 

The pitch of a worm or screw is usually given in a pitch of worm. 
way different from the pitch of a gear, viz: in num- 
ber of threads to one inch of the length of the worm or 
screw. Thus, to say a worm is 2 pitch may mean 2 
threads to the inch, or that the worm makes two turns 
to advance the thread one inch. But a worm may be 
double- threaded, triple- threaded and so on; hence to 
avoid misunderstanding, it is better always to call the 
advance of the worm thread the lead. Thus, a worm Lead of a 
thread that advances one inch in one turn we call one-inch 
lead in one turn. A single-thread worm 4 turns to 1" 
is X" lead. We apply the term pitch, that is the circular 
pitch, to the actual distance between the threads or teeth, 
as in previous chapters. In single-thread worms the 
lead and the pitch are alike. In making a worm and 
wheel a given number of threads to one inch, we divide 

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Fig. 40 

WORM AND WORM WHEEL 

THE THREAD OF WORM IS LEFT-HAND; 'WORM IS SINGLE-THREADED 



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BROWN & SHARPE MFG. CO. 

V by the number of threads to one inch, and the quotient 
is the circular pitch. Thus, the wheel in Fig. 41 is }4" 
Linear Pitch, eircular pitch. Linear pitch expresses exactly what is 
meant by circular pitch. Linear pitch has the 
advantage of being an exact use of language when applied 
to worms and racks. 

The number of threads to one inch linear, is the recipro- 
cal of the linear pitch. Thus, in the above example there 
are 2 threads to 1" as 2 is the reciprocal of }4" the linear 
pitch. We should say of a double- threaded worm 
advancing 1'' in 1 1/3 turns that: 

Lead - y^" or .75''. Linear Pitch or P' = 3/8'' or .375". 

Multiply 3.1416 by the number of threads to one inch, 
and the product will be the diametral pitch of the worm 
wheel. 

11/3 turns per 1" double-threaded = 2 2/3 threads 
per inch. 

2 2/3x3.1416-8.3776 times the diametral pitch or P. 

See Table of Tooth Parts. 
Drawing of To makc drawing of worm and wheel we obtain data 

Wormand • • ^ •, i 

Worm Wheel, as m circular pitch. 

1. Draw centre line AO and upon it space off the 
distance ab equal to the diameter of pitch circle. 

2. On each side of these two points lay off the dis- 
tance s, or the usual addendum = ^^^ as be and bd. 

3. From c lay off the distance cO equal to the radius 
of the worm. The diameter of a worm is generally four 
or five times the circular pitch. 

4. Lay off the distances eg and de each equal to /, 
or the usual clearance at bottom of tooth space. 

5. Through c and e draw circles about O. These 
represent the whole diameter of worm and the diameter 
at bottom of worm thread. 

6. Draw hO and iO at an angle of 30° to 45° with AO. 
These lines give width of face of worm wheel. 

7. Through g and d draw arcs about O, ending in„ 
hO and iO. 



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Teeth of 
Wheels fin- 



BROWN & SHARPE MFG. CO. 

This operation repeated at a completes the outline 
of worm wheel. For 32 teeth and more, the addendum 
diameter, or D, should be taken at the throat or smallest 
diameter of wheel, as in Fig. 41. Measure sketch for 
whole diameter of wheel blank. 

The foregoing instructions and sketch are for cases 
where the teeth of the wheels are finished with a hob. i^^ed with Hob 

A hob is shown in Fig. 42, being a steel piece threaded ^°^- 
with a tool of the same angle as the tool that threads 
the worm, the end of the tool being .3354 of the linear 
pitch; the hob is then grooved to make teeth for cutting, 
and hardened. 

The whole diameter of hob should be at least 2/, or 
twice the clearance larger than the worm. In our relieved ^°^ 
hobs the diameter is made about .005'' to .010" larger 
for small sizes to allow for wear. The outer corners of the 
hob thread can be rounded down as far as the clearance 
distance. The width at top of the hob thread before 
rounding should be .3095 of the linear, or circular pitch = 
.3095P'. The whole depth of thread is thus the ordinary 
working depth plus the clearance = D'' +/• The diameter 
at the bottom of the hob thread should be 2/+ .005'' to 
.010" larger for small sizes than the diameter at bottom of 
worm thread. In both this diameter and the outside 
diameter an allowance up to .03" or .04" can be made 
when hobs are of large size. For thread tool and worm 
thread see end of Chapter IV. 



Proportions of 




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BROWN & SHARPE MFG. CO. 



How to use 
the Hob. 



Universal 
Milling Ma- 
chine used in 
Hobbing. 



Why a Wheel 
is Hobbed. 



Worm Wheel 
Blanks with 
Less than 30 
Teeth. 



Interference 
of Thread and 
Flank. 



Example. 



Special Forms 
of Teeth. 



In the absence of a special worm gear cutting machine 
the teeth of the wheel are first cut as nearly to the finished 
form as practicable; the hob and worm wheel are mounted 
upon shafts and hob placed in mesh, it is then rotated 
and dropped deeper into the wheel until the teeth are 
finished. The hob generally drives the worm wheel 
during this operation. The universal milling machine 
is convenient for doing this work; with it the distance 
between axes of worm and wheel can be noted. In making 
wheels in quantities it is better to have a machine in 
which the work spindle is driven by gearing, so that the 
hob can cut the teeth from the solid without gashing. 
The object of hobbing a wheel is to get more bearing sur- 
face of the teeth upon worm thread. The worm wheels. 
Figs. 40 and 50, were hobbed. 

If we make the diameter of a worm wheel blank, that 
is to have less than 30 teeth, by the common rules for 
sizing blanks, and finish the teeth with a hob, we shall 
find the flanks of teeth near the bottom to be undercut 
or hollowing. This is caused by the interference spoken 
of in Chapter VI. Thirty teeth was there given as a 
limit, which will be right when teeth are made to circle 
arcs. With pressure angle 14>^°, and rack- teeth with 
usual addendum, this interference of rack-teeth with 
flanks of gear teeth begins at 31 teeth (31^ geometrically), 
and interfere with nearly the whole flank in a wheel of 
12 teeth. 

In Fig. 43 the blank for worm wheel of 12 teeth was 
sized by the same rule as given for Fig. 41. The wheel 
and worm are sectioned to show shape of teeth at the 
mid-plane of wheel. The flanks of teeth are undercut 
by the hob. The worm thread does not have a good 
bearing on flanks inside of A, the bearing being that of 
a corner against a surface. 

In Fig. 44 the blank for wheel was sized so that pitch- 
circle comes midway between outermost part of teeth 
and innermost point obtained by worm thread. 



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BROWN & SHARPE MFG. CO. 




Fig. 43 



^vrcii^/Rc/.^ 




'^M^' 





^..n-f 




Fig. 44 



Fig. 45 



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BROWN & SHARPE MFG. CO. 

This rule for sizing worm wheel blanks has been in 
use to some extent. The hob has cut away flanks of 
teeth still more than in Fig. 43. The pitch circle in 
Fig. 44 is the same diameter as the pitch circle in Fig. 
43. The same . hob was used for both wheels. The 
flanks in this wheel are so much undercut as to materially 
lessen the bearing surface of teeth and worm thread. 
Avowtd/'''"'" In Chapter VI the interference of teeth in high- 
numbered gears and racks with flanks of 12 teeth was 
remedied by rounding off the addenda. Although it 
would be more systematic to round off the threads of 
a worm, making them, like rack-teeth, to mesh with 
interchangeable gears, yet this has not generally been 
done, because it is easier to make a worm thread tool 
with straight sides. 

Instead of cutting away the addenda of worm thread, 
we can avoid the interference with flanks of wheels 
having less than 30 teeth by making wheel blanks larger. 

The flanks of wheel in Fig. 45 are not undercut, because 
the diameter of wheel is so large that there is hardly 
any tooth inside the pitch circle. The pitch circle in 
Fig. 45 is the same size as pitch circles in Figs. 43 and 44. 
This wheel was sized by the following rule: multiply the 
Th?oatto Avofd ^^^^^ diameter of the wheel by .937, and add to the product 
Interference. four timcs thc addcudum (45); the sum will be the 
diameter for the blank at the throat or small part. To get 
the whole diameter, make a sketch with diameter of 
throat to the foregoing rule and measure the sketch. 

It is impractical to hob a wheel of 12 to about 16 or 
18 teeth when blank is sized by this rule, unless the 
wheel is driven by independent mechanism and not by 
the hob. The diameter across the outermost parts of 
teeth, as at AB, is considerably less than the largest 
diameter of wheel before it was hobbed. 

In general it is well to size all blanks, as by page 78 
and Fig. 41, when the wheels are to be hobbed. The 
spaces can be cut the full depth, the cutter being 
dropped in. 

90 



BROWN & SHARPE MFG. CO. 

Fig. 46 shows a milling machine gashing the teeth of a 
worm wheel. 

In gashing the teeth the blank is dogged to the spiral 
head spindle, and the swivel table is swung to the required 
angle. The vertical feed is used and the teeth are indexed 
the same as in cutting a spur gear. Most of the stock is 
removed in gashing, enough only being left to allow the 
hob to take a light finishing cut. 

Fig. 47 shows the same wheel being hobbed. 

The work is set up practically the same as in the opera- 
tion of gashing the teeth, only the dog on the arbor is 
removed and the swivel table is set at zero. The worm 
wheel revolves freely on the centres, being rotated by 
the hob. 

The wheel can be hobbed to the right depth by using 
a steel rule at the back of the knee to measure a distance 
equal to the centre distance of the worm and wheel from a 
line marked "Centre", on the vertical slide to the top of 
the knee. This line on the vertical slide indicates the 
position of the top of the knee when the index centres 
are at the same height as the centre of the machine 
spindle. 

When worm wheels are not hobbed it is better to Blank Like a 
turn blanks like a spur wheel. Little is gained by having '^^'"' ^^^''^' 
wheels curved to fit worm unless teeth are finished with a 
hob. The teeth can be cut in a straight path diagonally 
across face of blank, to fit angle of worm thread, as in 
Figs. 48 and 51. 

In setting a cutter to gash a worm wheel. Figs. 49 and wheels for 
52, the angle is measured from the axis of the worm Machines'! 
wheel and the angle of the worm thread is, in conse- 
quence, measured from the perpendicular to the axis 
of the worm. See Chapters XVI and XIX. 

Some mechanics prefer to make dividing wheels in 
two parts, joined in a plane perpendicular to axis, hob 
teeth, then turn one part round upon the other, match 
teeth and fasten parts together in the new position, 
and hob again with a view to eliminate errors. With 

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BROWN & SHARPE MFG. CO. 




Fig. 46 
GASHING TEETH IN WORM WHEEL 



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BROWN & SHARPE MFG. CO. 




Fig. 47 
ROBBING TEETH IN WORM WHEEL 



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BROWN & SHARPE MFG. CO. 

an accurate cutting machine we have found wheels like 
Figs. 49 and 52, not hobbed, every way satisfactory. 
As to the different wheels, Figs 50, 51 and 52, when worm 

Diffe"renTstyies! IS in right position at the start, the lifetime of Fig. 50, 
under heavy and continuous work, will be the longest. 

Fig. 51 can be run in mesh with a gear or a rack as 
well as with a worm when made within the angular 
limits commonly required and is capable of lateral adjust- 
ment between the worm and wheel. Strictly, neither 
two gears made in this way, nor a gear and a rack would be 
mathematically exact, as they might bear at the sides 
of the gear or at the ends of the teeth only and not in 
the middle. At the start the contact of teeth in this 
wheel upon worm thread is in points only; yet such 
wheels have been many years successfully used in ele- 
vators. 

FiJfsMnJ'^ Rim^ Fig. 52 is a neat looking wheel. In gear cutting 
machines where the workman has occasion to turn the 
work spindle by hand, it is not so rough to take hold 
of as Figs. 50 and 51. The teeth are less liable to 
injury than the teeth of Figs. 50 and 51. 

The diameter of a worm has no necessary relation to 
the speed ratio of the worm to the worm wheel. The 
diameter of the worm can be chosen to suit any dis- 
tance between the worm shaft and the worm wheel 
shaft. It is unusual to have the diameter of the worm 
much less than four times the thread pitch or linear 
pitch but the worm can be of any larger diameter, five 
or ten times the linear pitch, if required. 

It is well to take off the outermost part of teeth in 
wheels (Figs. 40 and 50), as shown in these two figures. 
Limits for and not leave them sharp, as in Figs. 41 and 44. It is 
also well to round over the outer corners of the blanks 
for the wheels, Figs. 51 and 52. In ordering worms and 
worm wheels the centre distances should be given. If 
there can be any limit allowed in the centre distance 
it should be so stated. 



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BROWN & SHARPE MFG. CO. 




Fig. 48 

WORM WHEEL WITH TEETH CUT IN A STRAIGHT PATH DIAGONALLY 
ACROSS FACE. W^ORM IS DOUBLE-THREADED 



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BROWN & SHARPE MFG. CO. 




Fig. 49 / 

WORM AND WORM WHEEL FOR GEAR CUTTING MACHINE 



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BROWN & SHARPE MFG. CO. 



^^ 



Fig. 50 



Fig. 51 



Fig. 52 



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BROWN & SHARPE MFG. CO. 

By stating the limits that can be allowed, there may 
be a saving in the cost of work because time need not be 
wasted in trying to make work within narrower limits 
than are necessary. 

worm^and^of a Usually, in determining the length of a worm, the 

H°^' object is to make it just long enough to engage the teeth 

of the worm wheel in contact with it at one time. The 

length of the hob should be somewhat greater than that 

of the worm. 

The length of the worm varies with its diameter, 
the diameter and width of face of the worm wheel, pitch, 
pressure angle and helix angle, and to determine it for 
any particular case requires complicated calculations 
not within the scope of this treatise . 

For practical purposes the minimum length of worm 
can be determined by making a diagram (see Fig. 53) 
and measuring the length L of the addendum line repre- 
senting the outside diameter of the worm, at its inter- 
section with the throat circle of the worm wheel, or it 
may be calculated as follows: 

D = throat diameter of the worm wheel, D''= working 
depth of tooth, L = length of worm, V = length of hob. 



L = 2 Jd^'(D-D'0 V=L + 



If endwise movement of the worm relative to its wheel 
is required for adjustment or traverse the amount of 
such movement should also be added to the length of 
worm, but need not be added to the length of the hob. 

For a 30-tooth worm wheel of the form of Figs. 48 
and 49, we can have only about three threads in con- 
tact and a hob four threads long, like Fig. 42, is long 
enough. 

From the diagram, Fig. 54, which is similar to Fig. 7, 
we can tell approximately the number of threads that 
can bear. Let the worm move to the right and the 
action begins at C and ends at A', C being the point 

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BROWN & SHARPE MFG. CO. 




i Addendum Line 
Of Worm 



Fig. 53 




PITCH LINE 



Fig. 54 



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BROWN & SHARPE MFG. CO. 

where the line CD intersects the addendum circle of 
the gear and A' being the point where the line would 
intersect the addendum line of the worm. 

A short worm can be used in a large wheel by having 
the hob a little longer than the worm. 




GASHING TEETH OF HOB 



Hobs with 
Relieved Teeth. 



Grinding Hobs 
for Accuracy. 



Hobs of any size are made with the teeth relieved the 
same as gear cutters, the faces of which may be ground 
without changing the form of the teeth. They are 
made with a precision screw so that the pitch of the 
thread is accurate before hardening. 

When assured accuracy is desired for hardened hobs it 
can be obtained by grinding. 



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BROWN & SHARPE MFG. CO. 



CHAPTER XIV 



Sizing Gears When the Distance Between Centres 

and the Ratios of Speeds are Fixed — General 

Remarks — Width of Face of Spur Gears 

— Speed of Gear Cutters 



Let us suppose that we have two shafts 14'' apart, , centre d is - 

•^•^ r- 7 tance and Ratio 



centre to centre, and wish to connect them by gears so 
that they will have speed ratio 6 to 1. We add the 6 
and 1 together, and divide W by the sum and get 2'' 
for a quotient; this 2'', multiplied by 6, ^ives us the 
radius of pitch circle of large wheel=12''. In the same 
manner we get 2" as radius of pitch circle of small wheel. 
Doubling the radius of each gear, we obtain 24" and 4'' 
as the pitch diameters of the two wheels. The two num- 
bers that form a ratio are called the terms of the ratio. 
We have now the rule for obtaining pitch circle diameter 
of two wheels of a given ratio to connect shafts a given 
distance apart: 

Divide the centre distance by the sum of the terms of the 
ratio; find the product of twice the quotient by each term 
separately, and the two products will be the pitch diameters 
of the two wheels. 

It is well to give special attention to learning the 
rules for sizing blanks and teeth; these are much oftener 
needed than the method of forming tooth outlines. 

A blank l}4" diameter is to have 16 teeth; what will 
the pitch be? What will be the diameter of the pitch 
circle? See Chapter V. 

A good practice will be to compute a table of tooth 
parts. The work can be compared with the tables 
pages 178-181. 

In computing it is well to take tt to more than four 
places, TT to nine places=3. 141592653. ^ to nineplaces= 
.318309886. 

101 



fixed. 



Rule for Di- 
ameter of Pitch 
Circles. 



BROWN & SHARPE MFG. CO. 

Gearing"" ' "" Thcrc is HO such thing as pure rolling contact in teeth 
of wheels; they always rub, and, in time, will wear them- 
selves out of shape and may become noisy. 

Bevel gears, when correctly formed, run smoother 
than spur gears of same diameter and pitch, because 
the teeth continue in contact longer than the teeth of 
spur gears. For this reason annular gears run smoother 
than either bevel or spur gears. 

Sometimes gears have to be cut a little deeper than 
designed, in order to run easily on their shafts. If 
any departure is made in ratio of pitch diameters it is 
better to have the driving gear the larger, that is, cut 
the follower smaller. For wheels coarser than eight 
diametral pitch (8P), it is generally better to cut twice 
around, when accurate work is wanted, also for large 
wheels, as the expansion of parts from heat often causes 
inaccurate work when cut but once around. There is 
not so much trouble from heat in plain or web gears as 
in arm gears. 
G™fS;es.^^'''' ^^^ width of facc of cast iron gears can, for general 
use, be made 2>^ times the linear pitch. 

In small gears or pinions this width is often exceeded. 

The outer corners of spur gears may be rounded off 
for convenience in handling. This can be provided 
for when turning the blank. 
Speed of Gear Thc spccd of gcar cuttcrs is subject to so many con- 

Cutters. . :: . „ . - . 

ditions that definite rules cannot be given. 

Carbon cutters can be run from 60 to 70 feet per 
minute in cast iron and from 30 to 40 feet per minute 
in machinery steel. 

High speed steel cutters can be run from 80 to 125 
feet per minute in cast iron and from 65 to 100 feet per 
minute in machinery steel. 
Speed in In brass the speed of gear cutters can be twice as 
fast as in cast iron. Clockmakers and those making 
a specialty of brass gears exceed this rate even. A 12P 
cutter has been run 1200 turns a minute in bronze. A 
32P cutter has been run 7000 turns a minute in soft brass. 

102 



BROWN & SHARPE MFG. CO. 



The most desirable rate of feed varies widely under 
different conditions, while slight changes have so marked 
an effect on the cost and quality of the product that no 
exact rules can be given. 

The best method is to start with a given feed, then 
increase it until the gear blank will stand no more or the 
economical limit of the cutter is reached, and then use 
this or a very slightly slower feed for all similar work. 
One way to increase the production when cutting cast iron 
is to use an exhaust back of the cutter to carry away the 
chips and to keep the cutter cool. This will allow for 
materially higher speed and for the cutting of a much 
greater number of teeth without re-sharpening. 

As most of the cutters used for manufacturing are 
of high speed steel, the following table will give useful 
data for the feed of these cutters in cutting cast iron 
and low carbon steel. 



Rate of Feed. 



Use of A: 
Exhaust. 



Feed of High Speed Steel Cutters 



Diametral Pch 


2 


2i 


3 


4 


5 


6 


7 


8 


10 


12 


16 


Feed, 
Inches 


Cast I. 3| 


31 


4 


^ 


^ 


5 


6 


6 


7 


8 


9 


per 
Minute 


Steel 11 


n 


2 


2h 


2h 


3 


4 


4 


4i 


5 


6 



This table is based on finishing the gears in one cut. 
Whether this is permissible or not will depend on many 
things, such as the hardness of the material, size and 
shape or stiffness of the blank and the quality of finish 
desired. 

The matter of keeping cutters sharp is so important 
that it has sometimes been found best to have the work- 
man grind them at stated times, and not wait until he 
can see that the cutters are dull. Thus, have him grind 
every two hours or after cutting a stated number of 
gears. Cutters of the style that can be ground upon their 
tooth faces without changing form are rapidly destroyed 
if allowed to run after they are dull. Cutters are oftener 



Keep Cutters 
Sharp. 



103 



BROWN & SHARPE MFG. CO. 

wasted by trying to cut with them when they are dull 
than by too much grinding. Grind the faces radial with a 
free cutting wheel. Do not let the wheel become glazed, 
as this will draw the temper of the cutter. 

In Chapter VI was given a series of cutters for cut- 
ting gears having 12 teeth and more. Thus, it was 
there implied that any gear of same pitch, having 135 
teeth, 136 teeth, and so on up to the largest gears, and 
also, a rack, could be cut with one cutter. If this cut- 
ter is 4P, we would cut with it all 4P gears, having 135 
teeth or more, and we would also cut with it a 4P rack. 
Now, instead of always referring to a cutter by the num- 
ber of teeth in gears it is designed to cut, it has been 
found convenient to designate it by a letter or by a 
number. Thus, we call a cutter of 4P, made to cut gears 
135 teeth to a rack, inclusive. No. 1, 4P. 
We have adopted numbers for designating involute 
cutteS!"*^ ^^^'" S^^^ cutters as in the following table: 

No. 1 will cut wheels from 135 teeth to a rack inclusive. 

2 '' 

3 " 

4 - 

5 " 

6 '' 

7 " 

8 " 
By this plan it takes eight cutters to cut all gears 

having twelve teeth and over, of any one pitch. 

Thus, it takes eight cutters to cut all involute 4P 
gears having twelve teeth and more. It takes eight 
other cutters to cut all involute gears of 5P, having 
12 teeth and more. A No. 8, 5P cutter cuts only 5P 
gears having 12 and 13 teeth. A No. 6, lOP cutter 
cuts only lOP gears having 17, 18, 19 and 20 teeth. 
On each cutter is stamped the number of teeth at the 
limits of its range, as well as the number of the cutter. 
The number of the cutter relates only to the number 
of teeth in gears that the cutter is made for. 



55 ' 


' 134 teeth 


35 " 54 '' 


26 ' 


' 34 " 


21 ' 


' 25 " 


17 ' 


' 20 '' 


14 ' 


' 16 " 


12 ' 


' 13 " 



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BROWN & SHARPE MFG. CO. 



In ordering cutters for involute spur gears two things 
must be given : 

1. Either the number of teeth to be cut in the gear or the 
number of the cutter, as given in the foregoing table. 

2. Either the pitch of the gear or the diameter and number 
of teeth to be cut in the gear. 

li 2b teeth are to be cut in a 6P involute gear, the 
cutter will be No. 5, 6P, which cuts all 6P gears from 
21 to 25 teeth inclusive. If it is desired to cut gears 
from 15 to 25 teeth, three cutters will be needed. No. 
5, No. 6 and No. 7 of the pitch required. If the pitch 
is 8 and gears 15 to 25 teeth are to be cut, the cutters 
should be No. 5, 8P, No. 6, 8P, and No. 7, 8P. 

For each pitch of epicycloidal, or double-curve gears, 
24 cutters are made. In coarse pitch gears, the varia- 
tion in the shape of spaces between gears of consecu- 
tive numbered teeth is greater than in fine pitch gears. 

A set of cutters for each pitch to consist of so large 
a number as 24, was established for the reason that 
double-curve teeth were formerly preferred in coarse 
pitch gears. The tendency now, however, is to use the 
involute form in all cases. 

Our double-curve cutters have a guide shoulder on each 
side for the depth to cut. When this shoulder just reaches 
the periphery of the blank the depth is right. The marks 
which these shoulders make on the blank, should be as nar- 
row as can be seen, when the blanks are sized right . 

Double-curve gear cutters are designated by letters 
instead of by numbers; this is to avoid confusion in 
ordering. 

Following is the list of epicycloidal 
gear cutters: 

Cutter A cuts 12 teeth. Cutter M cuts 

- N 
O 
P 

" Q 

R 



B 


" 13 " 


" C 


- 14 " 


D 


" 15 " 


'' E 


" 16 - 


- F 


" 17 " 



ial or 


double-curve 


Its 27 


to 29 teeth. 


" 30 


'' 33 " 


" 34 


.. 37 - 


" 38 


.. 42 " 


" 43 


- 49 " 


" 50 


'' 59 " 



How to order 
Involute Cut- 
ters. 



Epicycloidal 
or D o u b 1 e - 
curve Cutters. 



Table of Epi- 
cycloidal or 
Double-curve 
Gear Cutters. 



105 



BROWN & SHARPE MFG. CO. 

Cutter G cuts 18 teeth. Cutter S cuts 60 to 74 teeth. 



" H ' 


' 19 " 




T 


'' 75 " 99 " 


.. I ' 


' 20 " 




U 


" 100 " 149 '^ 


'' J '' 21 to 22 




V 


'' 150 '' 249 '' 


" K ' 


' 23 to 24 




w 


'' 250 '' Rack. 


'' L ' 


' 24 to 26 




X 


" Rack. 



A cutter that cuts more than one gear is made of 
proper form for the smallest gear in its range. Thus, 
cutter J for 21 to 22 teeth is correct for 21 teeth; cutter 
S for 60 to 74 teeth is correct for 60 teeth, and so on. 
E^frj^^dofdli I^ ordering epicycloid al gear cutters designate the 
Cutters. letter of the cutter as in the foregoing table, also either 

give the pitch or give data that will enable us to deter- 
mine the pitch, the same as directed for involute cutters. 

More care is required in making and adjusting epi- 
cycloidal gears than in making involute gears. 
How to order In Ordering bevel gear cutters the following data 

Bevel Gear o o o 

Cutters. must be given: 

1. The number of teeth in each gear. 

2. Either the pitch of gears or the largest pitch diameter 

of each gear; see Fig. 18. 

3. The length of tooth face. 

If the shafts are not to run at right angles, it should 
be so stated, and the angle given. Involute cutters 
only are used for cutting bevel gears. No attempt 
should be made to cut epicycloidal tooth bevel gears 
with rotary disk cutters. 
How to order In ordcrlug worm wheel cutters, three things must 

Worm Gear ° 

Cutters. be given: 

1. Number of teeth in the wheel. 

2. Pitch of the worm; see Chapter XII I. 

3. Whole diameter of worm. 

In any order connected with gears or gear cutters, 
when the word * 'diameter" occurs, we usually under- 
stand that the pitch diameter is meant. When the 
whole diameter of a gear is meant it should be plainly 
written. Care in giving an order often saves the delay 

106 



BROWN & SHARPE MFG. CO. 

of asking for further instructions. An order for one gear 
cutter to cut from 25 to 30 teeth cannot be filled, because 
it takes two cutters of involute form to cut from 25 to 30 
teeth, and three cutters of epicycloidal form to cut 
from 25 to 30 teeth. 

In ordering, sheet zinc is convenient to sketch gears xempfetl ^""^ 
upon, and also for making templets. Before making 
sketch, it is well to give the zinc a dark coating with 
the following mixture: — dissolve 1 ounce of sulphate of 
copper (blue vitriol) in about 4 ounces of water, and add 
about one-half teaspoonful of nitric acid. Apply a thin 
coating with a piece of waste. 

This mixture will give a thin coating of copper to 
iron or steel, but the work should then be rubbed dry. 
Care should be taken not to leave the mixture where it 
is not wanted, as it rusts iron and steel. 

We have sometimes been asked why gears are noisy. ■nSS^gLvJ.'''' 
Not many questions can be asked us to which we can 
give a less definite answer than to the question why 
gears are noisy. 

We can indicate only some of the causes that may 
make gears noisy. When gears are cut too deep, which 
is more often the case rather than not deep enough, 
considerable noise results, especially if the driving gear is 
at fault. Cutting gears off centre may result in gea'rs 
being noisy in one direction when they may run quietly 
in the other direction. Another cause may be the centre 
distance, which if not right, allows the gears to mesh 
too tightly or run too loosely. Shafts that are not 
parallel or when the frame of the machine being of such 
a form as to give off sound vibrations are frequently 
found to be the cause of noisy gearing. 

There are numerous other causes for noisy gears and 
it is sometimes very difficult to tell what they may be, 
even after we have examined the gears in question. 



107 



BROWN & SHARPE MFG. CO. 

CHAPTER XV 

Spiral Gears^ — ^Calculations for Lead of Spirals 



spSl^Gear.°^ When the teeth of a gear are cut, not in a straight 
path, Hke a spur gear, but in a hehcal or screw-Hke path, 
the gear is called, technically a twisted or screw gear, 
but more generally among mechanics, a spiral gear. 
A distinction is sometimes made between a screw gear 
and a twisted gear. In twisted gears the pitch surfaces 
roll upon each other, exactly like spur gears, the axes 
being parallel, the same as in Fig. 1. In screw gears 
there is an end movement, or slipping of the pitch sur- 
faces upon each other, the axes not being parallel. In 
screw gearing the action is analogous to a screw and nut, 
one gear driving another by the end movement of its 
tooth path. This is readily seen in the case of a worm 
and worm wheel, when the axes are at right angles, as the 
movement of wheel is then wholly due to the end move- 
ment of worm thread. But, as we make the axes of 
gears more nearly parallel, they may still be screw gears, 
but the distinction is not so readily seen. 

Unless otherwise stated, the shafts of screw gears are 
at right angles, as at A and B, Fig. 56. 

The same gear may be used in a train of screw gears 
or in a train of twisted gears. Thus, B, as it relates to 
A, may be called a screw gear; but in connection with 
C, the same gear, B, may be called a twisted gear. These 
distinctions are not usually made, and we call all helical 
or screw-like gears spiral gears. 
Direction of Whcu two cxtcmal spiral gears run together, with 

eSn?i li* Axes' their axcs parallel, the teeth of the gears must have 
opposite hand spirals. 

Thus, in Fig. 56 the gear B has right-hand spiral teeth, 
and the gear C has left-hand spiral teeth. When the 
axes of two spiral gears are at right angles, both gears 

108 



BROWN & SHARPE MFG. CO. 




Fig. 55 
RACKS AND GEARS 




^ -^-iiMIMiiiiiiiiiiiilBliiiilliii^^ 



Fig. 56 
SPIRAL GEARING 

109 



BROWN & SHARPE MFG. CO. 

must have the same hand spiral teeth. A and B, Fig. 
56, have right-hand spiral teeth. If both gears A and B 
had left-hand spiral teeth, the relative direction in which 
they turn would be reversed. Fig. 57 shows in diagrama- 
tic form the hand and direction of revolution of spiral 
gears. 
Spiral Lead. ^^le Spiral Icad or lead of spiral is the distance the 
spiral advances in one turn on the pitch line. A cylinder 
or gear cut with spiral grooves is merely a screw of coarse 
pitch or long lead; that is, a spiral is a coarse lead screw, 
and a screw is a fine lead spiral. 

Since the introduction and extensive use of the uni- 
versal milling machine, it has become customary to 
call any screw cut in the milling machine a spiral. The 
spiral lead is given as so many inches to one turn. Thus, 
a cylinder having a spiral groove that advances six 
inches to one turn, is said to have a six inch spiral. 

In screws the pitch is often given as so many turns 
to one inch. Thus, a screw of }4" lead is said to be 2 
turns to the inch. The reciprocal expression is not 
much used with spirals. For example, it would not 
be convenient to speak of a spiral of 6'' lead, as } turns 
to one inch. 

The calculations for spirals are made from the func-. 
tions of a right angle triangle. 
shewing "Jiatlre ^^^ ^^^^ papcr a right angle triangle, one side of the 
rii^^^''' °^ ^p^- right angle 6" and the other side of the right aingle 2" 
long. Make a cylinder 6" in circumference. It will be 
remembered (Chapter II) that the circumference of a 
cyUnder, multiplied by .3183, equals the diameter; 
6''X.3183 = 1.9098'\ Wrap the paper triangle around 
the cyclinder, letting the 2" side be parallel to the axis, 
the 6" side perpendicular to the axis and reaching around 
the cylinder. The hypotheneuse now forms a helix 
or screw-like line, called a spiral. Fasten the paper 
triangle thus wrapped around. See Fig. 58. 

If we now turn this cylinder ABCD one turn in the 
direction of the arrow, the spiral will advance from O to E. 
This advance is the lead of the spiral. 

110 



BROWN & SHARPE MFG. CO. 




Fig. 57 
LEFT-HAND SPIRAL GEAR 



DIAGRAMS SHOWING DIRECTIONS OF REVOLUTIONS OF SPIRAL GEARS 





RIGHT-HAND 



LEFT-HAND 



111 



BROWN & SHARPE MFG. CO. 




Fig. 58 



Rules for cal- 
culating the 
parts of 
ral. 



spi- 



The angle EOF, which the spiral makes with the 
axis EO, is the angle of the spiral. This angle is found 
as in Chapter I. The circumference of the cylinder 
corresponds to the side opposite the angle. The pitch 
of the spiral corresponds to the side adjacent the angle. 
Hence the rule for angle of spiral : 

Divide the circumference of the cylinder or spiral by 
the number of inches of spiral to one turn, and the quotient 
will be the tangent of angle of spiral. For an explana- 
tion of the tangent and other functions of triangles, see 
Chapters XXII-XXIII. 

When the angle of spiral and circumference are given, 
to find the lead: 

Divide the circumference by the tangent of angle, and 
the quotient will be the lead of the spiral. 

When the angle of spiral and the lead or pitch of 
spiral are given, to find the circumference: 

Multiply the tangent of angle by the lead, and the product 
will be the circumference. 

When applying calculations to spiral gears the angle 
is reckoned at the pitch circumference and not at the 
outer or addendum circle. 



112 



BROWN & SHARPE MFG. CO. 

It will be seen that when two spirals of different dia- 
meters have the same lead the spiral of less diahieter 
will have the smaller angle. Thu$ in Fig. 58 if the paper 
triangle had been 4'' long instead of &' the diameter of 
the cylinder would have been 1.27'' and the angle of the 
spiral would have been only 63^ degrees. 



113 



BROWN & SHARPE MFG. CO. 



CHAPTER XVI 



Examples in Calculation of the Lead of Spiral 
Angle of Spiral — Circumference of Spiral 
Gears — A Few Hints on Cutting 



The rules for calculating the circumference of spiral 
gears, angle and the lead of spiral are the same as in 
Chapter XXII, for the tangent and angle of a right 
angle triangle. In Chapter XV, the word ''circum- 
ference" is substituted for ''side opposite," and the 
words "lead of spiral" are substituted for "side adjacent." 
rait^'lith^reler- When two Spiral gears are in mesh the angle of spiral 
orihaFts.^"^'^ should be the same in one gear as in the other, in order 
to have the shafts parallel and the teeth work properly 
together. When two gears both have right-hand spiral 
teeth, or both have left-hand spiral teeth, the angle of 
their shafts will be equal to the sum of the angles of their 
spirals. But when two gears have different hand spirals 
the angle of their shafts will be equal to the difference of 
their angles of spirals. Thus, in Fig. 56 the gears A and B 
both have right-hand spirals. The angle of both spirals is 
45°, their sum is 90°, or their axes are at right angles. But 
C has a left-hand spiral of 45°. Hence, as the difference 
between angles of spirals of B and C is 0, their axes 
are parallel. 

If two 45° gears of the same diameter have the same 
number of teeth the lead of the spiral will be alike in 
both gears: — if one gear has more teeth than the other 
the lead of spiral in the larger gear should be longer 
in the same ratio. Thus, if one of these gears has 50 
rais^^of '"differ- tccth, aud thc othcr has 25 teeth, the lead of spiral in 
ent diameters. .^^^ 50-tooth gcar shouM bc twlcc as long as that of 
the 25-tooth gear. Of course, the diameter of pitch 
circle should be twice as large in the 50-tooth as in the 
25-tooth gear. 

114 



BROWN & SHARPE MFG. CO. 

In spirals where the angle is 45° the circumference 
is the same as the spiral lead, because the tangent of 
45° is 1. 

Sometimes the circumference is varied to suit a pitch ciTcim1e?ence 
that can be cut on the machine and retain the angle *° ^""'^ ^ ^'^''■'^'• 
required. This would apply to cutting rolls for mak- 
ing diamond shaped impressions where the diameter 
of the roll is not a matter of importance. 

When two gears are to run together in a given velocity 
ratio, it is well first to select spirals that the machine will 
cut of the same ratio, and calculate the numbers of teeth 
and angle to correspond. This will often save con- 
siderable time in figuring. 

The calculations for spiral gears present no special 
difficulties, but sometimes a little ingenuity is required 
to make work conform to the machine and to such cutters 
as may be in stock. 

Let it be required to make two spiral gears to run 
with a ratio of 4 to 1, the distance between centres to 
be 3.125'' (Si/g"), the axes to be parallel. 

By rule given in Chapter XIV, we find the diameters 
of pitch circles will be 5" and 1^4"- Let us take a spiral 
of 48'' lead for the large gear, and a spiral of 12" lead for 
the small gear. The circumference of the 5" pitch circle 
is 15.70796". Dividing the circumference by the lead 
of the spiral, we have '-^^ = .3272" for tangent of angle 
of spiral. In the table the nearest angle to tangent, 
.3272", is 18°7'. 

As before stated, the angle of the teeth in the small 
gear will be the same as the angle of teeth or spiral in 
the large gear. 

. Now, this rule gives the angle at the pitch surface ^ difference 
only. Upon looking at a small screw of coarse pitch, ind"bittom*°Jf 
it will be seen that the angle at bottom of the thread ^p^^^^ Grooves, 
is not so great as the angle at top of thread; that is, 
the thread at bottom is nearer parallel to the centre 
line than that at the top. 



115 



BROWN & SHARPE MFG. CO. 



Example i n 
calculation of 
Lead of Spiral. 



This will be seen in Fig. 59, where AO is the centre 
line; bj shows direction of bottom of thread, and dg 
shows direction of top of thread. The angle Afb is 
less than the angle Agd. The difference of angle being 
due to the warped nature of a screw thread. 

A cylinder 2'' diameter is to have spiral grooves 20° 
with the centre line of cylinder; what will be the lead 
of spiral? The circumference is 6.2832". The tangent 
of 20° is .36397. Dividing the circumference by the 
tangent of angle, we obtain Up = 17.26"+ for lead of 
spiral. 




Fig. 59 

In Chapter XIII, it is stated that, when gashing 
the teeth of a worm wheel, the angle of the teeth across 
the face is measured from the line parallel to the axis 
of the wheel. 

To obtain this angle from the worm, divide the lead 
by the pitch circumference of the worm, and the quo- 
tient will be the tangent of the angle that the thread 
imakes with a plane perpendicular to the axis. 



116 



BROWN & SHARPE MFG. CO. 



CHAPTER XVII 



Normal Pitch of Spiral Gears — Curvature of Pitch 
Surface — Form of Cutters 



A Normal to a curve is a line perpendicular to the Normal to 

f. Curve. 

tangent at the point of tangency. 




Fig. 60 

In Fig. 60, the line BC is tangent to the arc DEF, 
.and the line AEO, being perpendicular to the tangent 
at E the point of tangency, is a normal to the arc. 

Fig. 61 is a representation of the pitch surface of a 
spiral gear. A'D'C is the circular pitch. ADC is the 
same circular pitch seen upon the periphery of a wheel. 
Let AD be a tooth and DC a space. Now, to cut this 
space DC, the path of cutting is along the dotted line ab. 
By mere inspection, we can see that the shortest distance 
between two teeth along the pitch surface is not the dis- 
tance ADC. 

Let the line AEB be perpendicular to the sides of 
teeth upon the pitch surface. A continuation of this 
line, perpendicular to all the teeth, is called the normal 
helix. The line AEB, reaching over a tooth and a space 
along the normal helix, is called the normal pitch or 
the normal linear pitch. 

117 



BROWN & SHARPE MFG. CO. 




Fig. 61 



118 



mal Pitch. 



BROWN & SHARPE MFG. CO. 

The normal pitch of a spiral gear is then: — the shortest formal pitch. 
distance between the centres of two consecutive teeth measured 
along the pitch surface. 

In spur gears the normal pitch and circular pitch 
are alike. In the rack DD, Fig. 55, the linear pitch and 
normal pitch are alike. 

From the foregoing it will be seen that, if we should spSai^ceai.'' "^ 
cut the space DC with a cutter, the thickness of which 
at the pitch line is equal to one-half the circular pitch, 
as in spur wheels, the space would be too wide, and 
the teeth would be too thin. Hence, spiral gears should 
be cut with thinner cutters than spur gears of the same 
circular pitch. 

The angle CAB is equal to the angle of the spiral. 
The line AEB corresponds to the cosine of the angle 
CAB. Hence the rule:— multiply the cosine of angle 
of spiral by the circular pitch and the product will be the to find ^ox- 
normal pitch. One-half the normal pitch is the proper 
thickness of cutter at the pitch line. 

If the normal pitch and the angle are known, divide 
the normal pitch by the cosine of the angle and the quo- 
tient will be the circular pitch: 

This may be required in a case of a spiral pinion run- 
ning in a rack. The perpendicular to the side of the 
rack is taken as the line from which to calculate angle 
of teeth. That is, this line would correspond to the 
axial line in a spiral gear; and, when the axis of the 
gear is at right angles to the rack, the angle of the teeth 
with the side of the rack is obtained by subtracting this 
angle from 90°. 

The angle of the rack teeth with the side of the rack 
can also be obtained by remembering that the cosine 
of the angle of spiral is the sine of the angle of the teeth 
with the side of the rack. 

The addendum and working depth of tooth should 
correspond to the normal pitch, and not to the circular 
pitch. Thus, if the norma] pitch is 12 diametral, the 
addendum should be //', the thickness .1309'', and so 

119 



BROWN & SHARPE MFG. CO. 

on. The diameter of pitch circle of a spiral gear is 
calculated from the diametral pitch. Thus a gear of 
30 teeth lOP would be ?>" pitch diameter. 

But if the normal pitch is 12 diametral pitch, the 
blank will be 3 -—;" diameter instead of 3~'^ 
orma i c jt is cvldent that with a given pitch diameter and 



vanes, 



number of teeth the normal pitch varies with the angle 
of spiral. The cutter should be for the normal pitch. 
In designing spiral gears, it is well first to look over 
list of cutters on hand, and see whether there are cutters 
to which the gears can be made to conform. This may 
avoid the necessity of getting a new cutter, or of changing 
both drawing and gears after they are under way. To 
do this, the problem is worked the reverse of the fore- 
going; that is: 
To make An- First calculatc to the next finer pitch cutter than 

f onf^'irm^To wouM bc rcQulred for the diametral pitch. 

Cutters given. ^^^ ^^ ^^^^^ ^^^ cxamplc, two gcars 10 pitch and 30 

teeth, spiral and axes parallel. Let the next finer cutter 
be for 12 pitch gears. The first thing is to find the 
angle that will make the normal pitch .2618'', when the 
circular pitch is .3142''. See Table of Tooth Parts. This 
means (Fig. 61) that the line ADC will be .3142" when 
AEB is .2618". Dividing .2618" by .3142" (see Chapter 
XV), we obtain the cosine of the angle CAB, which 
is also the angle of the spiral, '^^j^ = .8333. 

The same quotient comes by dividing 10 by 12, -\^ = 
.8333 + ; that is, divide one pitch by the other, the 
larger number being the divisor. Looking in the table, 
we find the angle corresponding to the cosine .8333 is 
33°34'. We now want to find the pitch of spiral that 
will give angle of 33°34' on the pitch surface of the wheel, 
3" diameter. Dividing the circumference by the tangent 
of angle, we obtain the pitch of spiral (see Chapter XVI). 
The circumference is 9.4248". The tangent of 33°34' 
is .66356, ^|f|| = 14.20; and we have for our spiral 
14.20" lead. 



120 



BROWN & SHARPE MFG. CO. 



When the machine is not arranged for the exact pitch 
of spiral wanted, it is generally well enough to take the 
next nearest spiral. A half of an inch more or less in a 
spiral 10'' pitch or more would hardly be noticed in angle 
of teeth. It is generally better to take the next longer spiral 
and cut enough deeper to bring centre distances right. 
When two gears of the same size are in mesh with their 
axes parallel, a change of angle of teeth or spiral makes 
no difference in the correct meshing of the teeth. 

But when gears of different size are in mesh, due 
regard must be had to the spirals being in pitch, pro- 
portional to their angular velocities (see Chapter XVI). 

We come now to the curvature of cutters for spiral 
gears; that is, their shape as to whether a cutter is made 
to cut 12 teeth or 100 teeth. A cutter that is right 
to cut a spur gear 3" diameter, may not be right for a 
spiral gear 3" diameter. To find the curvature of cutter, 
fit a templet to the blank along the line of the normal 
helix, as AEB, Fig. 61, letting the templet reach over 
about one normal pitch. The curvature of this templet 
will be nearer a straight line than an arc of the adden- 
dum circle. Now find the diameter of a circle that will 
approximately fit this templet, and consider this circle 
as the addendum circle of a gear for which we are to 
select a cutter, reckoning the gear as of a pitch the same 
as the normal pitch. 



When exact 
Pitch cannot be 
cut. 




Spiral Gears 
of Different 
Sizes of Mesh, 



Shape of Cut- 
ter. 



BROWN & SHARPE MFG. CO. 

Thus, in Fig. 62, suppose the templet fits a circle 
3}^" diameter, if the normal pitch is 12 to one inch, dia- 
metral, the cutter required is for 12P and 40 teeth. 
The curvature of the templet will not be quite circular, 
but is sufficiently near for practical purposes. Strictly, 
a flat templet cannot be made to coincide with the normal 
helix for any distance whatever, but any greater refine- 
ment than we have suggested can hardly be carried out 
in a workshop. 

This applies more to an end cutter, for a disk cutter 
may have the right shape for a tooth space and still 
round off the teeth too much on account of the warped 
nature of the teeth. 

The number of the cutter required may be calculated 
by the formula: — number of teeth for which cutter is to be 
selected = the number of teeth in the gear -^ by the cube 
of the cosine of the angle of teeth with the axis. Thus, in 
the example given on page 120, 30, the number of teeth -^ 
.8333^ = 52, the number of teeth for which the cutter 
should be selected. Referring to the table, page 104, it 
will be seen that a No. 3 cutter will be required. 

The difference, between normal pitch and linear or 
circular pitch is plainly seen in Figs. 55 and 56. 

The rack DD, Fig. 55, is of regular form, the depth 
of teeth being W of the circular pitch, nearly (.6866 of 
the pitch, accurately). If a section of a tooth in either 
of the gears be made square across the tooth, that is a 
normal section, the depth of the tooth will have the 
same relation to the thickness of the tooth as in the 
rack just named. 

But the teeth of spiral gears, looking at them upon 
the side of the gears, are thicker in proportion to their 
depth, as in Fig. 56. This difference is seen between 
the teeth of the two racks DD and EE, Fig. 55. In 
the rack DD we have 20 teeth, while in the rack EE 
we have but 14 teeth; yet each rack will run with each 
of the spiral gears A, B or C,^Fig. 56, but at different 
angles. 

122 



BROWN & SHARPE MFG. CO. 

The teeth of one rack will accurately fit the teeth of 
the other rack face to face, but the sides of one rack 
will then be at an angle of 45° with the sides of the other 
rack. At F is a guide for holding a rack in mesh with a 
gear. 

The reason the racks will each run with either of the 
three gears is because all the gears and racks have the 
same normal pitch. When the spiral gears are to run 
together they must both have the same normal pitch. 
Hence, two spiral gears may run correctly together 
though the circular pitch of one gear is not like the 
circular pitch of the other gear. 

If a rack is to run at any angle other than 90° with 
the axis of the gear it is well to determine the data from a 
diagram, as it is very difficult to figure the angles and 
sizes of the teeth without a sketch or diagram. 



123 



BROWN & SHARPE MFG. CO. 



CHAPTER XVIII 

Cutting Spiral Gears in a Universal Milling Machine 



Machine. 



A rotary disk cutter is generally preferable to a shank 
cutter or end mill on account of cutting faster and hold- 
ing its shape longer. In cutting spiral grooves, it is 
sometimes necessary to use an end mill on account of 
the warped character of the grooves, but it is very sel- 
dom necessary to use an end mill in cutting spiral gears. 
settinJ"of the Bcforc cutting into a blank it is well to make a slight 
trace of the spiral with the cutter, after the change 
gears are in place, to see whether the gears are correct. 
If the material of the gear blanks is quite expensive, it 
is a safe plan to make trial blanks of cast iron in order 
to prove the setting of the machine, before cutting into 
the expensive material. 

The cutting of spiral gears may develop some curi- 
ous facts to one that has not studied warped surfaces. 
The gears. Fig. 56, were cut with a planing tool in a 
shaper, the spiral gear mechanism of a universal mill- 
ing machine having been fastened upon the shaper. 
The tool was of the same form as the spaces in the rack 
DD, Fig. 55. All spiral gears of the same pitch can be 
cut in this manner with one tool. The nature of this 
cutting operation can be understood from a consideration 
of the meshing of straight side rack teeth with a spiral 
gear, as in Fig. 55. Spiral gears that run correctly 
with a rack, as in Fig. 55, will run correctly with each 
other when their axes are parallel, as at BC, Fig. 56 ; but it 
is not considered that they are quite correct, theoretically, 
to run together when the gears have the same hand spiral, 
and their axes are at right angles, as AB, Fig. 56, though 
they will run well enough practically. The operation of 
cutting spiral teeth with a planer tool is sometimes 
called planing the teeth. Planing is an accurate way of 

124 



BROWN & SHARPE MFG. CO. 





/> 






a 


c 





Fig. 63 



( 


> 

fi n 


/> 


<^ 


f) 




[| 11 


J 












— 








1 








\ 



Fig. 64 



125 



Data. 



BROWN & SHARPE MFG. CO. 

shaping teeth that are to mesh with rack teeth and for 
gears on parallel shafts; this method has been employed 
to cut spiral pinions that drive planer tables, but has not 
been found available for general use. 

It is convenient to have the data of spiral gears arranged 
as in the following table: 



Gear. 



No. of Teeth . 
Pitch Diameter. 
Outside Diameter 
Circular Pitch . 
Angle of Teeth with Axis 
Normal Circular Pitch 
Pitch of Cutter 
Addendum 5 
Thickness of Tooth t 

Whole Depth D'^+/ 
No. of Cutter . 
Exact Lead of Spiral 
Approximate Lead of Spiral 



Gears on Milling Machine to Cut Spiral 
Gear on Worm . . . 
1st Gear on Stud .... , 
2nd Gear on Stud . . . . 
Gear on Screw 



Pinion. 



A spiral of any angle to 45° can generally be cut in 
a universal milling machine without special attachments, 
the cutter being at the top of the work. The cutter is 
placed on the arbor in such position that it can reach 
the work centrally after the table is set to the angle of the 
spiral. In order to cut central, it is generally well enough 
to place the table, before setting it to the angle, so that 
the work centres will be central with the cutter, then 
swing the table and set it to the angle of the spiral. 



126 



BROWN & SHARPE MFG. CO. 



1^ 



.«^'<^;/>i^» 




Fig. 65 

USE OF VERTICAL SPINDLE MILLING ATTACHMENT 
IN CUTTING SPIRAL GEARS 



127 



BROWN & SHARPE MFG. CO. 

^_^centrai Set- p^^ ^^^^ accuratc work, it is safer to test the position 
of the centres after the table has been set to the angle. 

This can be done with a trial piece, Fig. 63, which 
is simply a round arbor with centre holes in the ends. 
It is mounted between the centres, and the knee is raised 
until the cutter sinks a small gash, as at A. This gash 
shows the position of the cutter; and if the gash is central 
with the trial piece, the cutter will be central with the 
work. If preferred, the arbor can be dogged to the work 
spindle; and the line BC drawn on the side of the arbor 
at the same height as the centres; the work spindle should 
then be turned quarter way round in order to bring 
the line at the top. The gash A can now be cut and its 
position determined with the line. 

In cutting small gears the arbor can be dogged to the 
work spindle; the distance between the gear blank and 
the dog should be enough to let the dog pass the cutter 
arbor without striking. 

A spiral gear is much more likely to slip in cutting 
than a spur gear. 

For gears more than three or four inches in diameter 
it is well to have a taper shank arbor held directly in 
the work spindle, as shown in Figs. 65 and 66; and for 
the heaviest work, the arbor can be drawn into the spin- 
dle with a screw in a threaded hole in the end of the 
shank. 

After cutting a space the work can be dropped away 
from the cutter, in order to avoid scratching it when 
coming back for another cut. Some workmen prefer 
not to drop the work away, but to stop the cutter and 
turn it to a position in which its teeth will not touch 
the work. To make sure of finding a place in the cut- 
ter that will not scratch, a tooth has sqmetimes been 
taken out of the cutter, but this is not recommended. 
The safest plan is to drop the work away. 
thfnfs".^'^^^*^'' In cutting spiral gears of greater angle than 45°, a 
vertical spindle milling attachment is available, as 
shown in Figs. 65 and 66. 

128 



BROWN & SHARPE MFG. CO. 




Fig. 66 

USE OF VERTICAL SPINDLE MILLING ATTACHMENT 
IN CUTTING SPIRAL GEARS 



129 



BROWN & SHARPE MFG. CO. 

In Fig. 65 the cutter is at 90° with the work spindle 
when the table is set to 0, so that the proper angle at 
which the table should be set, is the difference between 
the angle of the spiral and 90°. Thus, to cut a 70° 
spiral, we subtract 70° from 90°, and the remainder, 
20°, is the angle to set the table. In cutting on the 
top, Fig. 65, the attachment is set to 0. 

In Fig. QQ the cutter is at the side of the work; the 
table is set to 0, and the attachment is set to the differ- 
ence between 90° and the required angle of spiral. 
tinf "^^ ^°'' ^^*' In setting the cutter central it is convenient to have a 
small knee as at K, Fig. 64. A line is drawn upon the 
knee at the same height as at the centres. The cutter 
arbor is brought to the angle as just shown, and a gash 
is cut in the knee. When the gash is central with the 
line, the cutter will be central with the work. 

The cutter can be set to act upon either side of the 
gear to be cut, according as a right-hand or a left-hand 
spiral is wanted. The setting in Fig. QQ is for a right- 
hand spiral. 

If the gear blank were brought in front of the cutter, 
and the reversing gear set between two change gears, 
the machine would be set for a left-hand spiral. 

For coarser pitches than about 12P diametral, it is 
well to cut more than once around, the finishing cut 
being light so as to produce a smooth cut. 



130 



BROWN & SHARPE MFG. CO. 

CHAPTER XIX 

Spiral and Worm Gears — General Remarks 



The working of spiral gears, when their axes are parallel, spTaTfers. °^ 
is generally smoother than spur gears. A tooth does not 
strike along its whole face or length at once. Tooth 
contact first takes place at one side of the gear, passes 
across the face and ceases at the other side of the gear. 
This action tends to cover defects in shape of teeth and 
the adjustment of centres. 

Since the invention of machines for producing accu- 
rate epicycloidal and involute curves, however, it has not 
so often been found necessary to resort to spiral gears for 
smoothness of action. A greater range can be had in 
the adjustment of centres in spiral gears than in spur 
gears. The angle of the teeth should be enough, so 
that one pair of teeth will not part contact at one side 
of the gears until the next pair of teeth have met on the 
other side of the gears. When this is done the gears 
will be in mesh so long as the circumferences of their 
addendum circles intersect each other. This variation 
of centre distance is sometimes necessary in gears for rolls. 

Relative to spur and bevel gears in Chapter XIV, 
it was stated that all gears finally wore themselves out 
of shape and might become noisy. Spiral gears may be 
worn out of shape, but the smoothness of action can 
hardly be impaired so long as there are any teeth left. 
For every quantity of wear, of course, there will be an 
equal quantity of backlash, so that if gears have to be 
reversed the lost motion in spiral gears will be as much 
as in any gears, and may be more if there is end play of 
the shafts. In spiral gears there is end pressure upon the upon'^shafS^ol 
shafts, because of the screw-like action of the teeth. This ^^"^^ ^^^''^' 
end pressure is sometimes balanced by putting two gears 
upon each shaft, one of right and one of left-hand spiral. 

131 



Distinctive 
features of 



BROWN & SHARPE MFG. CO. 

The same result is obtained in solid cast gears by 
making the pattern in two parts — one right and one 
left-hand spiral. Such gears are colloquially called 
''herring-bone gears." 

In worm gears the axes are generally at right angles, 
or nearly so. The distinctive features of worm gearing 
may be stated as follows: 

The relative angular velocities do no depend upon 
the diameters of pitch cylinders, as in Chapter I. Thus, 
the worm in Chapter XIII, Fig. 40, can be any diameter — 
Worm Gearing. ^^^ ^^^^ ^^ ^^^ inchcs— without afifcctiug thc velocity 

of the worm wheel. Conversely if the axes are not 
parallel we can have a pair of spiral or worm gears of the 
same diameter, but of different numbers of teeth. The 
direction in which a worm wheel turns depends upon 
whether the worm has a right-hand or left-hand thread. 
When angles of axes of worm and worm wheel are oblique, 
there is a practical limit to the directional relation of 
the worm wheel. The rotation of the worm wheel is 
made by the end movement of the worm thread. 

The term worm and worm wheel, or worm gearing, 
is applied to cases where the worms are cut in a lathe, 
and the shapes of the threads or teeth, in axial section, 
are like a rack and the pitch is measured on a line parallel 
to the axis. The shape usually selected is like the rack 
for a single-curve or involute gear. See Chapter IV. 
Worms are sometimes cut in a milling machine. 

If the form of the teeth in a pair of worm gears is 
determined upon the normal helix, as in Chapter XVII, 
the gears are usually called spiral gears. 

If we let two cylinders touch each other, their axes 
being at right angles, the rotation of one cylinder will 
have no tendency to turn the other cylinder, as in 
Chapter I. 

The angle of a, worm thread can be calculated the 
same as the angle of teeth of a spiral gear; only, the 
angle of a worm thread is measured from a line or plane 
that is perpendicular to the axis of the worm. 

132 



BROWN & SHARPE MFG. CO. 

When a multiple-threaded worm is cut in a milling 
machine and the angle of the thread is less than 72° 
with the axis of the worm, it may be desirable to work 
by the normal pitch. The normal pitch can be obtained 
by multiplying the thread-pitch by the sine of the angle 
of the thread with the axis. 




WORM AND ^VORM AVHEEL IN POSITION FOR TESTING ON A 
BEVEL GEAR TESTING MACHINE 



133 



BROWN & SHARPE MFG. CO. 



TABLE No. 1 



No. OF 




Value 


OF "Y" 


Teeth 


For 


14|° Involute* 


For 20° Involute 


12 




.067 


,.078 


13 




.070 


.083 


14 




.072 


.088 


15 




.075 


.092 


16 




.077 


.094 


17 




.080 


.096 


18 




.083 


.098 


19 




.087 


.100 


20 




.090 


.102 


21 




.092 


.104 


23 




.094 


.106 


25 




.097 


.108 


27 




.100 


.111 


30 




.102 


.114 


34 




.104 


.118 


38 




.107 


.122 


43 




.110 


.126 


50 




.112 


.130 


60 




.114 


.134 


75 




.116 


.138 


100 




.118 


.142 


150 




.120 


.146 


300 




.122 


.150 


Rack 




.124 


.154 



* Originally given as 15° 



TABLE No. 2 

Safe Working Stress {^^) For Different Speeds 





Velocity of Pitch Line in Feet per Minute 


Material 





100 


200 


300 


600 


900 


1200 


1800 


2400 


Cast Iron 
Steel 


8000 
20000 


6850 
17000 


6000 
15000 


5350 
13300 


4000 
10000 


3200 
8000 


2650 
6650 


2000 
5000 


1600 
4000 



134 



BROWN & SHARPE MFG. CO. 

CHAPTER XX 

Strength of Gears 



There are in existence many rules for the strength of 
gear teeth, which have some merit when applied to the 
particular conditions for which they were designed. To 
establish a rule that would fit all conditions would be 
impracticable, and in adopting any rule we should see 
that as many factors as possible have been considered. 

The rule, or method, that has received the greatest 
recognition is that proposed by Wilfred Lewis and 
described by him in a paper read before the Engineers' 
Club of Philadelphia on October 15th, 1892, and published 
in the proceedings of the club, January, 1893. 

While the Lewis formula does not consider all the 
factors that enter into the problem, its almost universal 
acceptance proves quite conclusively its soundness for use 
in ordinary conditions of cut gearing. 

In this chapter we have adopted the Lewis formula with 
only such modifications as make it more adaptable for 
general use. 

The factors used are as follows : 

W = allowable load in pounds at the pitch line. 

S = allowable stress per sq. in. for static load (zero 

speed) for material of the gears and assumed to 

be 2/3 to ultimate strength for cast iron and 2/3 

the elastic limit of steel. 
P' = circular pitch of gear; distance from centre to 

centre of teeth on pitch line. 
F = Width of face of gear. 

Y = a factor for strength depending on pressure angle 

and number of teeth. (Table No. 1, page 134.) 

V = Velocity of pitch line in feet per minute; is the 

same for both gears of a pair. 

135 



BROWN & SHARPE MFG. CO. 

= a factor to modify or reduce the factor *'S" as 
the velocity *'V" increases from zero speed. 

HP = Horsepower. 

The equation for ''W" is: 

600XSPTY (1) 



W = 



and HP 



600 +V 

WV (2) 



33000 



Equation (1) is appHcable to any material and speed, 
but for average conditions the values given in Table 2, page 
134, can be used. When this is done the equation for *'W" 
is simplified and 

W-SPTY (3) 

WV 



and HP = 



33000 (2) 



Example: What is the allowable load '*W" in pounds 
and what horsepower can be transmitted by a 
pair of cast iron spur gears having 30T 6P1^" 
face running at 500 R.P.M? 

^^_ 5X3.1416X500 

12 
V = 654.5 ft. per minute. 

By referring to Table 2 under 600 F.P.M. we find the allow- 
able stress 

S = 4000 lbs. 

Now looking in Table 1 under 14>^° and opposite 30T we 
find that 



then 



Y = .102 

W = 4000 X.5236X 1.25 X. 102 (3) 

= 267 lbs. 

136 



BROWN & SHARPE MFG. CO. 

, ,T^ 267x654.5 ^^ , 

aad HP = = 5.3 horsepower (2) 

33000 

Now if these same sizes were to be made of a steel with 
an elastic limit of 60000 lbs. per square inch, we would 
proceed as follows: 

I X 60000 = 40000 = S for static load. 

.u w 600 X40000X. 5236 X 1.25 X. 102 (1) 

thenW = ^ 

600+654 

= 1277 lbs. 

and HP = ^^^'^X^^^ = 25 Horsepower (2) 

33000 

This is a case where the gears would have to be case- 
hardened and be thoroughly lubricated to transmit this 
power and give satisfactory wear. 

The values in Table 2 for "S" at zero speed are based on 
the original ones as given by Mr. Lewis, and are lower 
than necessary when the better grades of materials are 
used. In assuming higher values, however, it should be 
borne in mind that a point may be reached where excessive 
wear occurs in spite of the fact that the teeth are amply 
strong to withstand fracture. In such a case the question 
of wear becomes the limiting consideration and particular 
attention should be given to the combination of material 
of which the gears are made. 

For drives of small and medium size it is probable that 
gears of a high-grade casehardening steel, properly case- 
hardened, will transmit more power than those of any 
other available material. 

Where the gear is too large to caseharden readily, 
it is good practice to use a casehardened pinion with a 
gear of unhardened steel of not less than .40% carbon, 
and if the pinion can not be hardened, a steel .80% or more 
carbon could be used with the lower carbon gear. 

Both members are often made of cast iron and this is 
satisfactory when the work to be done is comparatively 
light, but if the ratio between the gear and pinion is large 

137 



BROWN & SHARPE MFG. CO. 

it is often advisable to use a steel pinion in order to 
equalize the strength and the wear. Both members of a 
drive should not be of unhardened low carbon steel unless 
the service is very light, as they are liable to seize and 
rough up. 

In calculating the strength of a pair of gears, each 
made of the same material, it is only necessary to consider 
the strength of the pinion, which is always the weaker 
member; but when made of different materials the 
strength of each member should be figured and the lesser 
of the two used in determining the strength of the drive. 

When a considerable amount of power is to be trans- 
mitted, it is highly beneficial to lubricate the tooth 
surfaces of the gears. If the speeds are moderate this 
can be satisfactorily accomplished by arranging one 
member so that it will dip into an oil bath, but where the 
speed is very high the centrifugal force tends to throw the 
oil off the teeth and under this condition it is advisable 
to arrange an oil pump so that oil can be applied to the 
teeth at the point of engagement. 

The effect of the speed or pitch line velocity upon the 
breaking strength of cast iron gears was investigated 
several years ago by Prof. Guido H. Marx of Leland 
Stanford University, and the results of the tests were 
published in the transactions of the American Society of 
Mechanical Engineers — Volume 34 — Page 1323, and 
Volume 37— Page 503. 

The gears used in these tests were made of cast iron 10 
diametral pitch, some of 14>^° pressure angle standard 
depth and others of 20° pressure angle and stub teeth. 

The general conclusions derived from these tests were 
that the Lewis formula, which assumes the entire load 
taken at approximately the end of a single tooth, does not 
give as great a strength- value as is warranted. This is 
because the contact between the teeth is near the pitch 
line when the load is carried on one tooth and when the 
contact is at the point of the tooth where there is a second 
tooth in contact thus dividing the load. 

138 



BROWN & SHARPE MFG. CO. 

A formula based on the results obtained from these 
tests gives lower values for the safe load at low speeds 
and higher values at high speeds than the Lewis formula. 

BEVEL GEARS 

To calculate the strength of bevel gears the following 

factors are used : 

W = allowable load at pitch line in pounds. 

S = allowable stress per sq. in. for static load (zero speed) 
for material of the gears and assumed to be 2/3 of the 
ultimate strength for cast iron and 2/3 the elastic 
limit for steel. 

Y = a factor for strength. 

P' = circular pitch. Distances from centre to centre of 

teeth at large end. 
F = face width of gears. 

V = velocity at pitch line (large end) in feet per minute; 

is the same for both gears of a pair. 
HP = horsepower. 
D' = pitch diameter at large end. 
d' = pitch diameter at small, end. 
N = actual number of teeth. 
n = formative number of teeth. The number of teeth 

for which the cutter is chosen when cutting bevel gears 

with a rotary cutter and for which the factor Y is 

chosen. See Fig. 23. 
a = edge angle of gear (pitch angle) see Fig. 77 Chapter 

XXIV. 

N (4) 

n = ^ 

COS. a 

W = SPFY§, (5) 

33000 (2) 

when S is taken from the table (2) . 

When other values for ''S" based on better material 
are taken, then — 

139 



BROWN & SHARPE MFG. CO. 



W = 600SPTY'd' 



600 +VD' 



HP= WV 
33000 



(6) 



(2) 



Before calculating the strength of a pair of bevel gears 
it is best to make use of a simple graphical layout as in 
Fig. 67 to get the diameter d' at the small end; this is 
easier than to calculate it. 



Of Gear 




Centreline 



Note. Symbols D'and A 
Have Sub Letters InThis 
Fig. Only In Using The 
Value For D'AudJl' Care 
MustBeTaken ToUseThe 
Proper One In The Formulas 



Pitch Diameter Of PmiON=D^ 



Fig. 67 



140 



BROWN & SHARPE MFG. CO. 

Example: 

What is the allowable load ''W" in pounds and what 
horsepower can be transmitted by a pair of bevel gears 
with 20 and 60T 6P lyi" face and 20^ pressure angle 
pinion running at 300 R.P.M. Pinion of steel and gear 
of cast iron. 

As **V" is the same for both gears we will base our 
calculation on this pinion. 

,, ,j 3.333x3.1416x300 ^.^ ,^ . ^ 

then V = = 262 ft. per mmute 

12 

"S" from Table 2 equals 13300 for steel and 5350 for iron. 

20 20 

n for pinion = = m this case ^,^^^ = 22 nearly 

COS. a .94869 

cjr\ pjr\ 

n for gear = = in this case ^^^^ =190 nearly 

COS. a .3162 

now taking the inside diameter d'p for pinion from Fig. 67 
we have 

W = 13300 X .5236X 1.5 X -104 X^|g ^^ 

W = 774 

774 V 262 
and HP = -^^^ = 6 Horsepower (2) 

For gear we have 

W= 5350 X. 5236 X 1.5 X. 146 X^^^^^ ® 

W = 440 

440 X 262 
then HP = -^355q- =3>^ Horsepower (2) 



141 



BROWN & SHARPE MFG. CO. 




142 



BROWN & SHARPS MFG. CO. 

CHAPTER XXI 

Standard Proportions for Spur Gears 



In order to ensure having gears of good proportion, 
and of sufficient strength without excessive weight with 
uniformity of appearance where the gears are used in 
sets, it is important to follow some regular system in 
proportioning the gears. 

The following formulas and tables have been used in our 
works for several years and are based on our experience 
in designing gears of this type. 

Standard Proportions for Spur Gears 



Formulas are for Gears 20P 4'' Diam. up to liP 12" Diam. 



A = .138D'(^^^+.28)+.67 



a = 



2A 



B = .4A 



.4a or ^? 



C = — 

D' = Pitch Diam, 



T=4 +.25 

G = (D"+/)+t+.06 



*K = 1.5H+it 
*L = 1.5H 
P = Diam. Pitch 



D" +/ = Full Depth of Tooth 
t = Thick, of Tooth 



Table Giving Value of (f- + .28) to Aid in Finding "A" 



Diam. Pitch 


1 

1.23 

10 

.38 


IK 
.915 

11 

.37 


2 

.755 

12 

.36 


2K 
.660 

13 

.354 


3 

.597 
14 
.350 


4 

.520 

15 

.345 


5 

.470 

16 

.340 


6 

.44 

17 

.337 


7 
.42 

18 
.334 


8 
.40 


9 


Value of(p +.28) .. 


39 






Diam. Pitch 


19 
.332 


20 


Value of (-^+.28) 


330 







*These proportions are suitable for general practice but must be varied to suit special 
cases: — Thus the face "F" should be less than given when used for light change gears 
and the hole "H" should vary to suit conditions but can usually be brought within one of 
the formulas (| +.5) ; (f +.5) or (4- +.5) . 



143 



BROWN & SHARPE MFG. CO. 

Rules and Table to Determine Whether Gears shall be Solid, Web or Arm 

It is specially important that, in a set to be used together, there should be consistency 
in making gears solid or with web or arms. 

The figures given are for the outside diameter of the largest gear in each style. 

Sizes larger than figures given in "Web" column are to be made with six arms, pro- 
portioned as per formulas above. 

The proportions for "Combination Gears" should be followed for ordinary cases, using 
the proportions for "Change Gears" only in cases where light service is required. 



Diam. Pitch. 
web ^^', 



Diam. Pitch . 











4 


5 


6 


7 


8 


9 


10 










5.50 


4.80 


4.50 


4.30 


4.25 


4.00 


3.80 










6.00 


5.00 


4.75 


4.57 


4.25 


4.00 


3.80 










9.00 


7.50 


7.30 


6.86 


6.75 


6.67 


5.80 










9.50 


8.00 


7.75 


7.00 


6.88 


6.75 


6.00 


12 




14 




16 


18 


20 










3.67 




3.29 




3.13 


3.00 


2.80 










3.67 




3.29 


















5.67 




5.30 




4.15 


4.55 


4.40 










5.75 




5.50 



















Make gear rims with a draft of 5° on inside. 

Make hubs with a taper of 3>^° on a side, except where finished, in which case do not 
taper hubs. 

{Round the corners on spur gear teeth as follows: — ^" Rad. to 20 P. Inc., ^" Rad. 
to 14 P. Inc., Ye" Rad. to 8 P. Inc., A" Rad. to 4 P. Inc., i" Rad. for coarser pitches. 

fMake diameter of hub = 1>^X Diam. of hole+><" except where heavy service is 
required, or large key ways or set screws are used, in which case make hub = If X diam. of 
hole-J-j", or reinforce hub over key way or around set screw. 



144 



BROWN & SHARPE MFG. CO. 



CHAPTER XXII 

Tangent of Arc and Angle 



We shall now show how to calculate some of the func- 
tions of a right angle triangle from a table of circular 
functions and the application of these calculations in some 
problems of gearing such as figuring the angles of bevel 
gears and in sizing blanks and cutting teeth of spiral 
gears, the selection of cutters for spiral gears, etc. 

A Function is a quantity that depends upon another 
quantity for its value. Thus the amount a workman 
earns is a function of the time he has worked and of his 
wages per hour. 




In any right angle triangle, OAB, we shall, for con- 
venience, call the two lines that form the right angle 
OAB the sides, instead of base and perpendicular. Thus 
OAB, being the right angle we call the line OA a side, and 
the line AB a side also. 

When we speak of the angle AOB, we call the line 
OA the side adjacent and the line AB the side opposite. 
When we are speaking of the angle ABO we call the line 
AB the side adjacent and the line OA the side opposite. 
The line connecting points OB is called the hypothenuse. 

In the following pages the definitions of circular func- 
tions are for angles smaller than 90°, and not strictly 
applicable to the reasoning employed in analytical 

145 



Functions of 
Right angle 
Triangle. 



Function de- 
fined. 



Right angle 
Triangle. 



Side adjacent. 



Hypothenuse. 



Tangent. 



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trigonometry, where we find expressions for angles of 
270°, 760°, etc. 

The Tangent of an arc is the line that touches it at 
one extremity and is terminated by a line drawn from 
the centre through the other extremity. The tangent 
is always outside the arc and is also perpendicular to 
the radius which meets it at the point of tangency. 




Fig. 69 



To find 
Degrees in 
Angle. 



the 
an 



Thus, in Fig. 69, the line AB is the tangent of the arc 
AC. The point of tangency is at A. 

An angle at the centre of a circle is measured by the 
arc intercepted by the sides of the angle. Hence the 
tangent AB of the arc AC is also the tangent of the 
angle AOB. 

In the tables of circular functions the radius of the 
arc is unity, or, in common practice, we take it as one 
inch. The radius OA being 1", if we know the length 
of the line or tangent AB we can, by looking in a table 
of tangents, find the number of degrees in the angle 
AOB. 

Thus, if AB is 2.25'' long, we find the angle AOB 
is QQ° 2' very nearly. That is, having found that 2.24956 
is the nearest number to 2.25 in the Table of Tangents 



146 



BROWN & SHARPE MFG. CO. 

at the end of this volume, we find the corresponding 
degrees of the angle in the column at the bottom of the 
table and the minutes to be added at the right-hand side 
of the table. 

Now, if we have a right angle triangle with an angle 
the same as OAB, but with OA two inches long, the 
line AB will also be twice as long as the tangent of angle 
AOB, as found in a table of tangents. 

Let us take a triangle with the side 0A = 5'' long, finl^'Thf" De- 
and the side AB = 8" long; what is the number of degrees ingfe" '" """ 
in the angle AOB? 

Dividing 8'' by 5 we find what would be the length 
of AB if OA was only V long. The quotient then would 
be the length of tangent when the radius is V long, 
as in the Table of Tangents. 8 divided by 5 is 1.6. The 
nearest tangent in the table is 1.6003 and the correspond- 
ing angle is 58°, which would be the angle AOB when 
AB is 8'' and the radius OA is 5'' very nearly. The 
difference in the angles for tangents 1.6003 and 1.6 could 
hardly be seen in practice. The side opposite the required 
acute angle corresponds to the tangent and the side 
adjacent corresponds to the radius. Hence the rule: 

To find the tangent of either acute angle in a right 
angle triangle: — divide the side opposite the angle by the 
side adjacent the angle and the quotient will be the tangent 
of the angle (see page 182). This rule should be com- 
mitted to memory. Having found the tangent of the 
angle, the angle can be taken from the Table of Tan- 
gents. 

The complement of an angle is the remainder after ofSnAngS!"* 
subtracting the angle from 90°. Thus 40° is the com- 
plement of 50°. 

The Cotangent of an angle is the tangent of the com- cotangent. 
plement of the angle. Thus, in Fig. 69, the line AB 
is the cotangent of AOE. In right angle triangles either 
acute angle is the complement of the other acute angle. 
Hence, if we know one acute angle, by subtracting this 
angle from 90° we get the other acute angle. As the arc 

147 



To find the 
Tangent. 



BROWN & SHARPE MFG. CO. 

approaches 90° the tangent becomes longer, and at 90° it is 
infinitely long. 

The sign of infinity is a. Tangent 90° = a. 
An^?e^^by''\he By a tablc of tangents, angles can be laid out upon 
JmSfe.'' Fig. ^o' sheet zinc, etc. This is often an advantage, as it is not 
convenient to lay protractor flat down so as to mark 
angles up to a sharp point. If we could lay off the 
length of a line exactly we could take tangents direct 
from table and obtain angle at once. It, however, is 
generally better to multiply the tangent by 5 or 10 
and make an enlarged triangle. If, then, there is a 
slight error in laying off length of lines it will not make 
so much difference with the angle. 

Let it be required to lay off an angle of 14°30'. By 
the table we find the tangent to be .25862. Multiply- 
ing .25862 by 5 we obtain, in the enlarged triangle, 
1.29310" as the length of side opposite the angle 14° 
30'. As we have made the side opposite five times as 
large, we must make the side adjacent five times as large, 
in order to keep angle the same. Hence, Fig. 70, draw 
the line AB 5'' long; perpendicular to this line at A draw 
the line AO 1.293'' long; now draw the line OB, and the 
angle ABO will be 14°30'. 

If special accuracy is required, the tangent can be 
multiplied by 10; the line AO will then be 2.586" long 
and the line AB 10" long. Remembering that the 
acute angles of a right angle triangle are the comple- 
ments of each other, we subtract 14°30' from 90° and 
obtain 75°30' as the angle of AOB. 

The reader will remember these angles as occurring 
in Chapter IV, and obtained approximately in a different 
way. A semicircle upon the line OB touching the extremi- 
ties O and B will just touch the right angle at A, and the 
line OB is four times as long as OA. 

Let it be required to turn a piece 4" long, \" diam- 
eter at small end, with a taper of 10° one side with the 
other; what will be the diameter of the piece at the 
large end? 

148 



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Fig. 70 



Fig. 71 



149 



BROWN & SHARPE MFG. CO. 

Dilmeter^''pf^*f A sectioii, Fig. 71, through the axis of this piece is 

5ece^ Mg! 7L ^he saiiie as if we added two right angle triangles, O 

AB and O'A'B', to a straight piece A'ABB^ V wide 

and 4'' long, the acute angles B and B^ being 5°, thus, 

making the sides OB and O'B'10° with each other. 

The tangent of 5° is .08749, which, multiplied by 
4'', gives .34996" as the length of each line, AO and 
A'O', to be added to V at the large end. Taking twice 
.34996'' and adding to V, we obtain 1.69992'' as the 
diameter of large end. 

This chapter must be thoroughly studied before taking 
up the next chapters. If once the memory becomes 
confused as to the tangent and sine of an angle, it will 
take much longer to get righted than it will to first care- 
fully learn to recognize the tangent of an angle at once. 

If one knows what the tangent is, one can tell better 
the functions that are not tangents, 
problems fnTri- ^0 solvc problcms in right angle triangles see 'Table 
angles. £qj. ^^le Solutlou of Right Angle Triangles," page 182. 



150 



BROWN & SHARPE MFG. CO. 

CHAPTER XXIII 

Sine and Cosine — Some of their Applications in 
Machine Construction 



The Sine of an arc is the line drawn from one extremity 
of the arc to the diameter passing through the other 
extremity, the line being perpendicular to the diameter. 

Another definition is: — the sine of an arc is the dis- 
tance of one extremity of the arc from the diameter, 
through the other extremity. 

The sine of an angle is the sine of the arc that measures 
the angle. 

In Fig. 72, AC is the sine of the arc BC, and of the 
angle BOC. It will be seen that the sine is always 
inside of the arc, and can never be longer than the radius. 
As the arc approaches 90°, the 
sine comes nearer to the radius, 
and at 90° the sine is equal to 1, or 
is the radius itself. From the defini- 
tion of a sine, the side AC, opposite 
the angle AOC, in any right angle 
triangle, is the sine of the angle AOC, 
when OC is the radius of the arc. 
Hence the rule: — in any right angle 
triangle, the side opposite either acute 
angle, divided by the hypothenuse, is equal to the sine of the 
angle. (See table, page 182). 

The quotient thus obtained is the length of side opposite 
the angle when the hypothenuse or radius is unity. The 
rule should be carefully committed to memory. 

A Chord is a straight line joining the extremities of 
an arc, and is twice as long as the sine of half the angle 
measured by the arc. Thus, in Fig. 72, the chord FC 
is twice as long as the sine AC. 




Fig. 72 



Sine of Arc 
and Angle. 



To find the 



Sine. 



Chord of an 
Arc. 



151 



BROWN & SHARPE MFG. CO. 



/ 


3 \ 




,''''""/^ 




"""v 






"s 


/* / 




s 


/ / 
/ / 

/ / 




s 
\ 






\ 


1 / \ 




\ 
\ 


/ X \^ 




\ 






\ 
\ 
\ 


\) 





■ 1 ' 

\ 1 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 


\ 




/ 

/ 


^v 




y 


\ 




y 


"-^. r 


■^ 


^^ 




.1. 



Fig. 73 



Example to 
find the Chord. 



Polygon. 



Let there be four holes equidistant about a circle 
?>" in diameter, Fig. 73; what is the shortest distance 
between two holes? This shortest distance is the chord 
AB, which is twice the sine of the angle COB. The 
angle AOB is one-quarter of the circle, and COB is one- 
eighth of the circle. 360°, divided by 8 = 45°, the angle 
COB. The sine of 45° is .70711, which multiplied by the 
radius 1.5", gives length CB in the circle, 3'' in diameter, 
as 1.06066''. Twice this length is the required distance 
AB = 2.1213". 

When a cylindrical piece is to be cut into any num- 
ber of sides, the foregoing operation can be applied to 
obtain the width of one side. A plane figure bounded 
by straight lines is called a polygon. 

152 



BROWN & SHARPE MFG. CO. 



Cosine. 



When the outside diameter and the number of sides of ierj?h of side. 
a regular polygon are given, to find the length of one of 
the sides: — divide 360° by twice the number of sides; multi- 
ply the sine of the quotient by the outer diameter, and the 
product will be the length of one of the sides. 

Multiplying by the diameter is the same as multi- 
plying by the radius, and that product again by 2. 

The Cosine of an angle is the sine of the complement 
of the angle. 

In Fig. 72, COD is the complement of the angle AOC; 
the line CE is the sine of COD, and hence is the cosine 
of BOC. The line OA is equal to CE. It is quite as well 
to remember the cosine as the part of the radius, from 
the centre that is cut off by the sine. Thus, the sine 
AC of the angle AOC cuts off the cosine OA. The line 
OA may be called the cosine because it is equal to the 
cosine CE. 

In any right angle triangle, the side adjacent either 
acute angle corresponds to the cosine when the hypothe- 
nuse is the radius of the arc that measures the angle; 
hence: — divide the side adjacent the acute angle by the 
hypothenuse, and the quotient will be the cosine of the angle. 

When a cylindrical piece is cut- into a polygon of any 
number of sides, a table of cosines can be used to obtain Length of 
the diameter across the sides. ^on"' °^ ^°^^^'' 



To find the 
Cosine. 




Rule for Di- 
ameter across 
"sides of a Poly- 
gon. 



BROWN & SHARPE MFG. CO. 

Let a cylinder, 2" diameter, Fig. 75, be cut six-sided; 
what is the diameter across the sides? 

The angle AOB, at the centre, occupied by one of 
these sides, is one-sixth of the circle, = 60°. The cosine 
of one-half this angle, 30°, is the line CO; twice this Hne 
is the diameter across the sides. The cosine of 30° is 
.86603, which, multiplied by 2, gives 1.73206^' as the 
diameter across the sides. 

Of course, if the radius is other than unity, the cosine 
should be multiplied by the radius, and the product 
again by 2, in order to get diameter across the sides; 
or what is the same thing, multiply the cosine by the 
whole diameter or the diameter across the corners. 

The rule for obtaining the diameter across sides of 
regular polygon, when the diameter across corners is 
given, will then be: — multiply the cosine of 360° divided 
by twice the number of sides, by the diameter across corners, 
and the product will be the diameter across sides. 

The Table of Sines and Cosines is arranged like the 
Table of Tangents and Cotangents as explained on 
page 146. 

C B 




Fig. 75 



To find the 
Diameter across 
corners of a 
Polygon. 



A six-sided piece is to be 1}4" across the sides; how 
large must a blank be turned before cutting the sides? 
Dividing 360° by twice the number of sides or 180° by 
the number of sides, we have 30°, which is the angle COB, 
Fig. 75. 



154 



BROWN & SHARPE MFG. CO. 

The radius of the six-sided piece is .1^" . Dividing 
.75'' by the cosine of 30° which will be found by the table 
to be .86603 gives .8660 the hypothenuse OB. .8660 X 2 = 
1.7320+ the required diameter of blank. 

Hence, in a regular polygon, when the diameter across 
sides and the number of sides are given, to find diameter 
across corners, divide the distance across sides by the 
cosine of 180 divided by the number of sides and the 
quotient will be the distance across corners. 



155 



BROWN & SHARPE MFG. CO. 




No. 13 AUTOMATIC GEAR CUTTING MACHINE 
FOR SPUR AND BEVEL GEARS 

Cuts Spur and bevel gears to 18^' diameter, 4^' face. 
Cast iron, 5 diametral pitch; steel, 6 diametral pitch. 



156 



BROWN & SHARPE MFG. CO. 



CHAPTER XXIV 



Application of Circular Functions — Whole Diameter 
of Bevel Gear Blanks — Angles of Bevel Gear Blanks 



The rules given in this chapter apply only to bevel 
gears having the centre angle c'Oi not greater than 90°. 

To avoid confusion we will illustrate one gear only. 
The same rules apply to all sizes of bevel gears. Fig. 
76 is the outline of a pinion 4P, 20 teeth, to mesh with 
a gear 28 teeth, shafts at right angles. For making 
sketch of bevel gears see Chapter X. 

In Fig. 76, the line Om'm is continued to the line ab. 
The angle c'Oi that the cone pitch line makes with the 
centre line may be called the centre angle. The centre 
angle c'Oi is equal to the angle of edge c'ic, c'i is the mgi 
side opposite the centre angle c'Oi, and c'O is the side 
adjacent the centre angle. c'/ = 2.5"; c'0 = 3.5". Divid- 
ing 2.b" by 3.5" we obtain .71428''+ as the tangent 
of c'Oi. In the table we find .71417 to be the nearest 
tangent, the corresponding angle being 35°32'. 35°32^ 
then, is the centre angle c'Oi and the angle of edge c'in, 
very nearly. 

When the axes of bevel gears are at right angles the 
angle of edge of one gear is the complement of angle 
of edge of the other gear. Subtracting, then, 35°32' 
from 90° we obtain 54°28' as the angle of edge of gear 
28 teeth, to mesh with gear 20 teeth. Fig. 76, from which 
we have the rule for obtaining centre angles when the 
axes of gears are at right angles. 

Divide the radius of the pinion by the radius of the gear 
and the quotient will be the tangent of centre angle of the 
pinion. 

Now subtract this centre angle from 90° and we have 
the centre angle of the gear. 



157 



Angle of 



BROWN & SHARPE MFG. CO. 




Fig. 76 
BEVEL GEAR DIAGRAM 



158 



BROWN & SHARPE MFG. CO. 

The same result is obtained by dividing the number of 
teeth in the pinion by the number of teeth in the gear; the 
quotient is the tangent of the centre angle. 

To obtain angle of face Om"c', the distance c'O becomes ^ngie of Face. 
the side opposite and the distance m"c' is the side adja- 
cent. 

The distance c'O is 3.5'', the radius of the 28-tooth 
bevel gear. The distance c'm" is by measurement 
2.82". 

Dividing 3^5 by 2.82 we obtain 1.2411 for tangent 
of angle of face Om"c'. The nearest tangent in the 
table is 1.24079 and the corresponding angle is 51°8'. 
To obtain cutting angle c'On" we divide the distance 
c'n" by c'O. By measurement c'^" is 2.2". Divid- 
ing 2.2 by 3.5 we obtain .62857 for tangent of cutting 
angle. The nearest corresponding angle in the table 
is 32°9'. 

The largest pitch diameter, kj, of a bevel gear, as in 
Fig. 77, is known the same as the pitch diameter of 
any spur gear. Now, if we know the distance ho or 
its equal aq, we can obtain the whole diameter of bevel 
gear blank by adding twice the distance bo to the largest 
pitch diameter. 

Twice the distance bo, or what is the same thing, Diameter in- 
the sum of aq and bo is called the diameter increment, 
because it is the amount by which we increase the largest 
pitch diameter to obtain the whole or outside diameter 
of bevel gear blanks. The distance bo can be calculated 
without measuring the diagram. 

The angle boj is equal to the angle of edge. 

The angle of edge, it will be remembered, is the angle 
formed by outer edge of blank or ends of teeth with the 
end of hub or a plane perpendicular to the axis of gear. 

The distance bo is equal to the cosine of angle of edge, 
multiplied by the distance jo. The distance jo is the 
addendum, as in previous chapters ( = 5). 

Hence the rule for obtaining the diameter increment 
of any bevel gear: — multiply the cosine of angle of edge 

159 



BROWN & SHARPE MFG. CO. 




Fig. 77 
DIAGRAM— BEVEL GEAR AND PINION 

160 



Outside Diam- 
eter. 



BROWN & SHARPE MFG. CO. 

by the working depth of teeth (D"), cind the product will be the 
diameter increment. 

By the method given on page 157, we find the angle 
of edge of gear (Fig. 77) is 56° 19'. The cosine of 56°19' 
is .55436, which, multipHed by 2/3", or the depth of the 
3P gear, gives the diameter increment of the bevel gear 
18 teeth, 3P meshing with pinion of 12 teeth. 2/3 of 
.55436 = .3696" (or .37", nearly). Adding the diameter 
increment, .37", to the largest pitch diameter of gear, 
&\ we have 6.37" as the outside diameter. 

In the same manner, the distance cd is half the dia- 
meter increment of the pinion. The angle cdk is equal 
to the centre angle of pinion, and when axes are at right 
angles is the complement of centre angle of gear. The 
centre angle of pinion is 33°40^ The cosine, multiplied 
by the working depth, gives .555" for diameter incre- 
ment of pinion, and we have 4.555" for outside diameter 
of pinion. 

In turning bevel gear blanks, it is sufficiently accu- 
rate to make the diameter to the nearest hundredth of 
an inch. 

The small angle oOj is called the angle increment. ^^^^^^ ^""^' 
When shafts are at right angles the face angle of one 
gear is equal to the centre angle of the other gear, minus 
the angle increment. 

Thus, the angle of face of gear (Fig. 77) is less than 
the centre angle DO^, or its equal Ojk by the angle oOj. 
That is, subtracting oOj from Ojk, the remainder will 
be the angle of face of gear. 

Subtracting the angle increment from the centre 
angle of gear, the remainder will be the cutting angle. 

The angle increment can be obtained by dividing 
oj, the side opposite, by O;', the side adjacent, thus finding 
the tangent as usual. 

The length of cone pitch line from the common centre, Length of 
O to j, can be found, without measuring diagram, by 
dividing the adjacent side OB of the triangle OB7 by the 
cosine of the centre angle; to find the length of cone pitch 

161 



BROWN & SHARPE MFG. CO. 

line, divide the side adjacent to the centre angle by the cosine 
of the centre angle. 

The length of the side adjacent equals the radius of the 
pinion 2" which divided by .55436 the cosine of 56°19' = 
3.6045'' the length of the line Oj. 

Dividing oj by Oj, we have for tangent .0924, and 
for angle increment 5° 17'. 

The angle increment can also be obtained by the 
following rule: 

Divide the sine of centre angle by half the number of 
teeth, and the quotient will be the tangent of increment angle. 

Subtracting the angle increment from centre angles 
of gear and pinion, we have respectively: 
Cutting angle of gear, 51° 2'. 
Cutting angle of pinion, 28°25'. 

Remembering that when the shafts are at right angles, 
the face angle of a gear is equal to the cutting angle of its 
mate (Chapter XI), we have: 

Face angle of gear, 28°25'. 
Face angle of pinion, 51°2'. 

It will be seen that both the whole diameter and the 
angles of bevel gears can be obtained without making 
a diagram. "Formulas in Gearing," published by us, 
contains extensive tables for bevel gearing, including tables 




Sine. 



BROWN & SHARPE MFG. CO. 

for diameter increment, angle of face and edge, etc., for 
bevel gears. 

In laying out angles, the following method may be AnSfe^^by''\he 
preferred, as it does away with the necessity of making 
a right angle: — draw a circle, ABO (Fig. 78), ten inches 
in diameter. Set the dividers to ten times the sine of 
the required angle, and point off this distance in the 
circumference as at AB. From any point O in the cir- 
cumference, draw the lines OA and OB. The angle 
AOB is the angle required. Thus, let the required angle 
be 12°. The sine of 12° is .20791, which, multiplied by 
10, gives 2.0791'', or 2^" nearly, for the distance AB. 

Any diameter of circle can be taken if we multiply the 
sine by the diameter, but 10" is very convenient, as all 
we have to do with the sine is to move the decimal point 
one place to the right. 

If either of the lines pass through the centre, then the 
two lines which do not pass through the centre will form a 
right angle. Thus, if OB passes through the centre then 
the two lines AB and AO will form a right angle at A. 



163 



BROWN & SHARPE MFG. CO. 

CHAPTER XXV 

Angle of Pressure 



In Fig. 79, let A be any flat disk lying upon a hori- 
zontal plane. Take any piece, B, with a square end, ab. 
Press against A with the piece B in the direction of the 
arrow. 





Fig. 79 



Fig. 80 



It is evident A will tend to move directly ahead of 
B in the normal line cd. Now (Fig. 80) let the piece 
B, at one corner /, touch the piece A. Move the piece 
B along the line de, in the direction of the arrow. 

It is evident that A will not now tend to move in 
the line de, but will tend to move in the direction of the 
normal cd. When one piece, not attached, presses against 
another, the tendency to move the second piece is in 
the direction of the normal, at the point of contact. 
Line of Press- TMs normal is called the line of pressure. The angle that 
this line makes with the path of the impelling piece, is 
called the angle of pressure. 

In Chapter IV, the lines BA and BA' are called lines 
of pressure. This means that if the gear drives the rack. 



164 



BROWN & SHARPE MFG. CO. 

the tendency to move the rack is not in the direction of 
pitch line of rack, but either in the direction BA or BA', as 
we turn the wheel to the left or to the right. 

The same law holds if the rack is moved in the direction 
of the pitch Hne; the tendency to move the wheel is not 
directly tangent to the pitch circle, as if driven by a belt, 
but in the direction of the line of pressure. Of course the 
rack and wheel do move in the paths prescribed by their 
connections with the framework, the wheel turning about 
its axis and the rack moving along its ways. This press- 
ure, not in a direct path of the moving piece, causes extra 
friction in all toothed gearing that cannot well be avoided. 

Although this pressure works out by the diagram, 
as we have shown, yet, in the actual gears, it is not at 
all certain that they will follow the law as stated, because 
of the friction of teeth among themselves. If the driver 
in a train of gears has no bearing upon its tooth-flank, 
we apprehend there will be but little tendency to press 
the shafts apart. 

The arc through which a wheel passes while one of 
its teeth is in contact is called the arc of action. 

Until within a few years, the base of a system of double- tem 
curve interchangeable gears was 12 teeth. It is now g 
15 teeth in the best- practice. (See Chapter VII) . 

The reason for this change was: the base, 15 teeth, 
gives less angle of pressure and longer arc of contact, 
and hence longer lifetime to gears. 



Arc of Action. 



Base of Sys- 
of I nter- 
changeable 
ears. 



165 



BROWN & SHARPE MFG. CO. 

CHAPTER XXVI 

Continued Fractions — Some Applications in 
Machine Construction 



Fractions. 



a Conti*Jrue°d A continucd fraction is one that has unity for its 

numerator, and for its denominator an entire number 

plus a fraction, which fraction has also unity for its 

numerator, and for its denominator an entire number 

plus a fraction, and thus in order. 

The expression, ^^f- 

s+r 

^ is called a continued fraction. 

By the use of continued fractions, we are enabled 

of^cpntinu'^d to find a fraction expressed in smaller numbers, that, 

for practical purposes, may be sufficiently near in 

value to another fraction expressed in large numbers. 

If we were required to cut a worm that would mesh 

with a gear 4 diametral pitch (4P), in a lathe having 

3 to 1-inch linear leading screw, we might, without 

continued fractions, have trouble in finding change 

gears, because the circular pitch corresponding to 4 

diametral pitch is expressed in large numbers: 4P = SqP'- 

This example will be considered farther on. For 

illustration, we will take a simpler example. 

What fraction expressed in smaller numbers is near- 
est in value to ^g ? Dividing the numerator and the 
denominator of a fraction by the same number does 
not change the value of the fraction. Dividing both 

1 

by 29, we have qi or, what is the same 

1 

thing expressed as a continued fraction, 5+j, . The 

29 
1 29 

continued fraction s+T is exactly equal to 145 • If 

29 

now, we reject the 29, the fraction -5 will be larger than 



Example 



r. ,. ^5 terms of 

Continued 
Fractions. 



166 



BROWN & SHARPE MFG. CO. 
1 

5+j^ , because the denominator has been diminished, 

29 

5 being less than 629. y is something near 146 expressed 
in smaller numbers than 29 for a numerator and 146 for a 

1 29 

denominator. Reducing -5 and 1^ to a common 
denominator, we have j = ^^ and ^ = ^o- Subtracting 
one from the other, we have 730, which is the difference 
between j and f|.. Thus, in thinking of §q as j, we 
have a pretty fair idea of its value. 
There are fourteen fractions with terms smaller than „ -^^^L^.^f °^ 



approximation. 



valent . 



29 and 146, which are nearer i^e than j is, such as 76' si 

28 

and so on to yTi- In this case by continued frac- 
tions we obtain only one approximation, namely -5, and 
any other approximations, as ^, gi, etc., we find by 

trial. It will be noted that all these approximations 

29 
are smaller in value than j^. There are cases, how- 
ever, in which we can, by continued fractions, obtain 
approximations both greater and less than the required 
fraction, and these will be the nearest possible approxi- 
mations that there can be in smaller terms than the 
given fraction. 

In the French metric system, a millimetre is equal ^,„Metric equi- 
to .03937 inch; what fraction in smaller terms expresses 
.03937'' nearly? .03937, in a vulgar fraction, is i^. 

Dividing both numerator and denominator by 3937, 

1 

we have 0^1575 . Rejecting from the denominator 

'^'-'3937 

of the new fraction, Iff, the fraction ^ gives us a 

pretty good idea of the value of .03937". If in the 

J 

expression, 2.5+1575, we divide both terms of the 

3937 
fraction ~ by 1575, the value will not be changed. 
Performing the division, we have l..^ 

2+787 
1575. 

We can now divide both terms of {§-^ by 787, without 
changing its value, and then substitute the new fraction 
for ^ in the continued fraction. 

167 



BROWN & SHARPE MFG. CO. 

Dividing again, and substituting, we have: 
1 

25+1 

2+1 
2 +1 
787 

as the continued fraction that is exactly equal to .03937. 
In performing the divisions, the work stands thus: 

3937)100000(25 

7874 

21260 
19685 
1575) 3937 (2 
3150 



787) 1575 (2 
1574 



1) 787 (787 
787 



That is, dividing the last divisor by the last remain- 
der, as in finding the greatest common divisor. The 
quotients become the denominators of the continued 
fraction, with unity for numerators. The denominators 
25, 2, and so on, are called incomplete quotients, since 
they are only the entire parts of each quotient. The 
first expression in the continued fraction is 2^ or .04 — 
a little larger than .03937. If, now, we take ^^» 
we shall come still nearer .03937. The expression 2^^ is 
merely stating that 1 is to be divided by 25 >^. To 
divide, we first reduce 25>^ to an improper fraction,^, 

and the expression becomes 51, or one divided by f. 

2 

To divide by a fraction, 'Invert the divisor, and proceed 
as in multiplication." We then have ^ as the next 
nearest fraction to .03937. ^ = .0392 + , which is smaller 
than .03937. To get still nearer, we take in the next 
part of the continued fraction, and have 



25+1 

2+1 
2. 



We can bring the value of this expression into a frac- 
tion, with only one number for its numerator and one 
number for its denominator, by performing the operations 
indicated, step by step, commencing at the last part of 
the continued fraction. Thus, 2 + >^, or lyi, is equal 

168 



BROWN & SHARPE MFG. CO. 

to -f. Stopping here, the continued fraction would 
become .,Vt, 

2o+l 



Now, 5^ equals f, and we have 25+2 . 25^ equals 
2 5 

127. „,.i ^:^..^: :„ i -\- t>v' •^.' _ 1 u,. 127 



-5-; substituting again, we have 127^ . Dividing 1 by 



5 » 



we have ^. ^j is the nearest fraction to .03937, unless 
we reduce the whole continued fraction ~ 



25+1 

2+1 

787, which 
would give us back the .03937 itself. —=.03937007, 
which is only 100000000 larger .03937. It is not often that 
an approximation will come so near as this. 

This ratio, 5 to 127, is used in cutting millimetre Ppcticai use 

01 the toregoing 

thread screws. If the leading screw of the lathe is Example. 
1 to one inch, the change gears will have the ratio of 
5 to 127: if 8 to one inch, the ratio will be 8 times as 
large, or 40 to 127; so that with leading screw 8 to inch, 
and change gears 40 and 127, we can cut millimetre threads 
near enough for practical purposes. 

The foregoing operations are more tedious in description 
than in use. The steps have been carefully noted, 
so that the reason for each step can be seen from rules 
of common arithmetic, the operations being merely 
reducing complex fractions. The reductions, ^, ^, jlj, 
etc., are called conver gents, because they come nearer 
and nearer to the required .03937. The operations 
can be shortened as follows : 

Let us find the fractions converging towards .7854" Example. 
the circular pitch of 4 diametral pitch, .7854 = ^0 J 



169 



BROWN & SHARPE MFG. CO. 



reducing to lowest terms, we 

the operation for the greatest common divisor: 



have ig. 



3927) 5000 (1 
3927 

1073) 3927 (3 
3219 
708) 1073 (1 
708 

365) 708 (1 
365 



343^ 365 (1 
343 



22) 343 (15 
22 
l23 
110 



13) 22 (1 
13 



9) 13 (1 



4) 9 (2 
_8 

1) 4 (4 
4 



Applying 



Rule 



Bringing the various incomplete quotients as denomi- 
nators in a continued fraction as before, we have : 

1 

1 + 1 



3 + 1 

1 + 1 



1 + 1 

1 + 1 



15 + 1 

1 + 1 

1 + 1 

2 + i 



Now arrange each partial quotient in a line, thus: 



1 1 1 15 



3 


4 


7 


11 


172 


183 


355 


893 


3927 


4 


5 


9 


14 


219 


233 


452 


1137 


5000 



Now place under the first incomplete quotient the 
first reduction or convergent y, which, of course, is 1; 
put under the next partial quotient the next reduction 
or convergent 1^3/3 or 1%, which becomes % . 

1 is larger than .7854, and ^ is less than .7854. 

Having made two reductions, as previously shown, we 
can shorten the operations by the following rule for next 
convergents: — multiply the numerator of the convergent 
just found by the denominator of the next term of the con- 

170 



BROWN & SHARPE MFG. CO. 

tinned fraction, or the next incomplete quotient, and add 
to the product the numerator of the preceding convergent: 
the sum will he the numerator of the next convergent. 

Proceed in the same way for the denominator, that is 
multiply the denominator of the convergent just found 
by the next incomplete quotient and add to the product 
the denominator of the preceding convergent; the sum 
will be the denominator of the next convergent. Con- 
tinue until the last convergent is the original fraction. 
Under each incomplete quotient or denominator from 
the continued fraction arranged in line, will be seen the 
corresponding convergent or reduction. The con- 
vergent H is the one commonly used in cutting racks 
4P. This is the same as calling the circumference of a 
circle f when the diameter is one (1); this is also the 
common ratio for cutting any rack. The equivalent 
decimal to ^ is .7857+, being about ^^ large. In 
three settings for rack teeth, this error would amount to 
about .OOr'. 

For a worm, this corresponds to n threads to V'\ now, 
with a leading screw of lathe 3 to I", we would want 
gears on the spindle and screw in a ratio of 33 to 14. 

Hence, a gear on the spindle with 66 teeth, and a gear 
on the 3 thread screw of 28 teeth, would enable us to cut 
a worm to fit a 4P gear. 

In "Formulas in Gearing", tables of factors and prime 
numbers are given which are of assistance in problems 
requiring the use of continued fractions. 



171 



BROWN & SHARPE MFG. CO. 

CHAPTER XXVII 

Squares and Square Roots 



To square a number, multiply it by itself. 

The expression 7^ indicates that 7 is to be squared. 
72 = 7 X 7 = 49, the square of 7. 
234' = 234 X 234 = 54756, the squaie of 234. 

To obtain the square root of a number proceed as 
follows: 

(1) Space the number into groups of two figures each 
both ways from the decimal point. 

(2) Find the greatest square in the first group on the 
left, and set its root on the right. Subtract the square 
thus found from the first group and to the remainder 
annex the two figures of the next following group for a 
dividend. 

(3) Double the root first found for a divisor and find 
how many times it is contained in the dividend exclusive 
of the right hand figure of the latter and set that quotient 
figure both in the quotient and divisor. 

Multiply the whole divisor by the last quotient figure 
and subtract the product from the dividend, bringing 
down the next group for a new dividend. 

(4) Repeat the process under Rule 3 and so on through 
all the periods to the last. 

Let us find" the square root of 54756. 



172 



BROWN & SHARPE MFG. CO. 



The expression \l 54756 indicates that the square 
root of 54756 is to be found. 







5 47 56 


234 Ans. 


2' 


^ = 4 




2X2=4— 


1 47 




Annex 3 
43 


1 29 




23X2 = 46— 


1856 




\nnex 4 






464 


1856 













Therefore x' 54756 = 234. 
Find the square root of .1968 



4' 


I 


19 68 
16 


.443+ Ans. 


4x2=8— 

Annex 4 

84 


3 68 
3 36 


44x2 = 8^ 
Annex 


3 


320( 
264^ 


) 


SI 


B3 


) 



551 



Therefore \L 1986 = .443 + 



173 



BROWN & SHARPE MFG. CO. 

TABLES 

TABLE GIVING CHORDAL THICKNESS OF GEAR TEETH 

(t'O AND DISTANCE FROM CHORD TO 

TOP OF TOOTH (s") 




When accurate measure- 
ments of gear teeth are 
required, it is necessary 
to work to the chordal 
thickness of tooth. This 
thickness, also the distance 
from the chord to the top 
of tooth, varies from the 
figures given in the ''Table of 
Tooth Parts" being greatest 
for gears with the fewest teeth. 

The table gives the chordal thick- 
ness of teeth and the distance from 
the chord to the top of tooth for 
gears of 1 diametral pitch. 

To obtain V and s'' for any 
diametral pitch divide the figures 
given in the table opposite the 
required number of teeth by the 
required diametral pitch. 

Example: — Find f' and s'' for a ' 

gear 5 diametral pitch, 23 teeth 
1.5696 -^5 = . 3139 = t'' 
1.0268-^5 = .2054 = s'' 

To obtain V and s'' for any circular pitch multiply the 
figures given in the table, opposite the required number 
of teeth by s (taking ''s" from the ''Table of Tooth 
Parts", pages 178 and 179). 

Example: — Find V and s'' for a ^'' circular pitch 
gear, 15 teeth. 

1. 5679 X. 2387 = . 3743 = t^ 
1.0411 X. 2387 = . 2485 = s^ 



174 



BROWN & SHARPE MFG. CO. 

CHORDAL THICKNESS OF GEAR TEETH FOR 1 DIAMETRAL PITCH 



NUMBER 
OF TEETH 


t" 


s" 


NUMBER 
OF TEETH 


t" 


s* 


NUMBER 
OF TEETH 


t" 


s" 














94 




5707 


1.0066 


6 




55^9 


1. 1022 


50 




5705 


1.0123 


95 




5707 


1.0065 


7 




55^^^ 


I.0S73 


51 




5706 


1.0121 


96 




5707 


1.0064 


8 




5607 


1.0769 


52 




5706 


1.0119 


97 




5707 


1.0064 


9 




5628 


1,0684 


53 




5706 


I.01I7 


98 




5707 


1.0063 


10 




5643 


I.0616 


54 




5706 


I.01I4 


99 




5707 


1.0062 


1 1 




5654 


1-0559 


55 




5706 


I.01I2 


100 




5707 


1. 0061 


12 




S'^^j 


1.0514 


56 




5706 


1. 01 10 


101 




5707 


1.0061 


13 




5670 


1.0474 


57 




5706 


1.0108 


102 




5707 


1.0060 


14 




5675 


1.0440 


58 




5706 


1.0106 


103 




5707 


1.0060 


15 




5679 


1. 04 1 1 


59 




5706 


I.OIO5 


104 




-5707 


1.0059 


16 




56S3 


1-0385 


60 




5706 


1.0102 


105 




5707 


1.0059 


17 




5686 


1.0362 


61 




5706 


I.OIOl 


106 




5707 


1.0058 


18 




.5688 


1.0342 


62 




5706 


I.OIOO 


107 




5707 


1.0058 


19 




5690 


1.0324 


63 




5706 


1.0098 


108 




5707 


1.0057 


20 




5692 


1.0308 


64 




5706 


1.0097 


109 




5707 


1.0057 


21 




5694 


1.0294 


65 




5706 


1.0095 


1 10 




5707 


1.0056 


22 




5695 


1.0281 


66 




5706 


1.0094 


1 i 1 




5707 


1.0056 


23 




5696 


1.0268 


67 




5706 


1.0092 


1 12 




5707 


1-0055 


24 




5697 


1.0257 


68 




5706 


1.0091 


1 13 




5707 


1-0055 


25 




5698 


1.0247 


69 




5707 


1.0090 


1 14 




5707 


1.0054 


26 




5698 


1.0237 


70 




5707 


1.0088 


1 15 




5707 


i.ooc;4 


27 




5699 


1.0228 


71 




5707 


1.0087 


1 16 




5707 


1-0053 


28 




5700 


1.0220 


72 




5707 


1.0086 


1 17 




5707 


1-0053 


29 




5700 


1. 02 1 3 


73 




5707 


1.0085 


1 18 




5707 


1-0053 


30 




5701 


1.0208 


74 




5707 


1.0084 


1 19 




5707 


1.0052 


31 




5701 


1.0199 


75 




5707 


1.0083 


120 




5707 


1.0052 


32 




5702 


1.0193 


76 




5707 


1.0081 


121 




5707 


1.0051 


33 




5702 


1.0187 


77 




5707 


1.0080 


122 




5707 


1. 0051 


34 




5702 


1.0181 


78 




5707 


1.0079 


123 




5707 


1.0050 


35 




5702 


1.0176 


79 




5707 


1.0078 


124 




5707 


1.0050 


36 




5703 


1. 0171 


80 




5707 


1.0077 


125 




5707 


1.0049 


37 




5703 


1.0167 


81 




5707 


1.0076 


126 




5707 


1.0049 


38 




5703 


1.0162 


82 




5707 


1.0075 


127 




5707 


1.0049 


39 




5704 


1.0158 


83 




5707 


1.0074 


128 




5707 


1.0048 


40 




5704 


1-0154 


84 




5707 


1.0074 


129 




5707 


1.0048 


41 




5704 


1.0150 


85 




5707 


1.0073 


130 




5707 


1.0047 


42 




5704 


1.0147 


86 




5707 


1.0072 


131 




5708 


1.0047 


43 




5705 


1.0143 


87 




5707 


1.0071 


132 




5708 


1.0047 


44 




5705 


1.0140 


88 




5707 


1.0070 


133 




5708 


1.0047 


45 




5705 


1.0137 


89 




5707 


1.0069 


134 




5708 


1.0046 


46 




5705 


1.0134 


90 




5707 


1.0068 


135 




5708 


1.0046 


47 




5705 


1.0131 


91 




5707 


1 .0068 


150 




5708 


1.0045 


48 




5705 


1.0129 


92 




5707 


1.0067 


250 




5708 


1.0025 


49 


1 


5705 


1.0126 


93 




5707 


1.0067 


Rack 




5708 


1.0000 



175 



BROWN & SHARPE MFG. CO. 

DIAMETRAL PITCH. 

"NUTTALL." 
Diametral Pitch is the Number of Teeth to Each Inch of the Pitch Diameter. 



To Get 



Havinj 



Rule. 



Formula. 



The Diametral 
Pitch. 



The Diametral 
Pitch. 



The Diametral 
Pitch. 



Pitch 

Diameter. 



Pitch 

Diametei 



Pitch 

Diametei 



Pitch 

Diameter. 



Outside 

Diameter 



Outside 

Diameter 



Outside 

Diameter, 



Outside 

Diameter 



Number of 

Teeth. 



Number of 

Teeth. 



Thickness 

of Tooth 



Addendum. 



Root. 



Workins 



Depth. 
Whole Depth. 

Clearance. 

Clearance. 



The Circular Pitch. 

The Pitch Diameter 
and the Number of 
Teetii .... 

The Outside Diame- 
ter and theNunibe 
of Teeth .... 

The Number of Teetli 
and the Diametral 
Pitch 

The Number of Teeth 
and Outside Diam- 
eter 

The Outside Diame- 
ter and the Diam- 
etral Pitch . . . 

Addendum and the 
Number of Teeth. 

The Number of Teeth 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Number of 
Teeth 

The Number of Teeth 
and Addendum . 

The Pitch Diameter 
and tlie Diametral 
Pitch 

The Outside Diame- 
ter and the Diame- 
tral Pitch . . . 



The Diametral Pitch 
The Diametral Pitch. 

The Diametral Pitch. 
The Diametral Pitch. 
The Diametral Pitch. 
The Diametral Pitch. 
Thickness of Tooth. 



Divide 3.1416 by the Circular Pitch 

Divide Number of Teeth by Pitch 
Diameter 



Divide Number of Teeth plus 2 by 
Outside Diameter 



Divide Numl)er of Teeth by the 
Diametral Pitch 

Divide the i)roduct of Outside 
Diameter and Numl)er of Teeth 
by Number of Teeth plus 2 

Subtract from the Outside Diame 
ter the quotient of 2 divided by 
the Diametral Pitch .... 

Multiply Addendum by the Num- 
ber of Teeth ....... 



Divide Number of Teeth plus 2 by 
the Diametral Pitch . . . . ' 



Add to the Pitch Diameter the 
quotient of 2 divided by the 
Diametral Pitch 

Divide the Number of Teeth ])lu 
2 by the quotient of Number of 
Teeth and by the Pitch Diameter 

Multiply the Numl)er of Teeth 
plus 2 by Addendum .... 

Multiply Pitch Diameter by the 
Diametral Pitch 



Multiply Outside Diameter by the 
Diametral Pitch and subtract 2. 



Divide 1.5708 by the Diametral 
Pitch 

Divide 1 by the Diametral Pitch, 
D' 
or s = -^ 

Divide 1.157 by the Diametral Pitch 



Divide 2 by the Diametral Pitch. 
Divide 2.157 by the Diametral Pitch 

Divide .157 by the Diametral Pitch 

Divide Thickness of Tooth at 
pitch line by 10 



.1416 





P' 


= 


N 
D' 




N+2 




D 


- 


N 

"P~ 




DN 



N+2 



D'= D — 



D'= 


sN 


D^ 


N+2 
P 


D = 




D = 


N+2 
N . 
D'' 


D = 


(N+2) s 



N, = D'P 

N = DP — 

1.5708 



t = - 



S+f: 



D"= 



1.157 



2.157 



f^ 



.157 



176 



BROWN & SHARPE MFG. CO. 



CIRCULAR PITCH. 

"NUTTALL." 

Circular Titch is the Distance from the Centre of One Tootli to the Centre of the 
Next Tooth, Measured along the Pitch Line. 



To Get 



The Circular 
Pitch, 



The Circular 
Pilch, 



The Circular 
Pitch, 



Pitch 

Diameter, 



Pitch 

Diameter, 



Pitch 

Diameter. 



Pitch 

Diameter, 



Outside 

Diameter 



Outside 

Diameter, 



Outside 

Diameter. 



Number of 

Teeth. 



Thickness 

of Tooth 



Addendum. 

Root. 

Workinjr 

Depth. 

Whole Depth. 

Clearance. 

Clearance. 



Havinf 



The Diametral Pitch. 

The Pitch Diameter 
and the Number of 
Teeth 

The Outside Diame- 
ter and the Number 
of Teeth „ . . . 

The Number of Teeth 
and the Cii'cular 
Pitch 

The Number of Teeth 
and the Outside Di- 
ameter .... 

The Outside Diame- 
ter and the Circular 
Pitch 

Addendum and the 
Number of Teeth. 

The Number of Teeth 
and the Circular 
Pitch 

The Pitch Diameter 
and the Circular 
Pitch ..... 

The llumber of Teeth 
r.nu t!ic Addendimi 

The Pitch Diameter 
and the Circular 
Pitch . . . 

The Circular Pitch. 
Tiie Circular Pitch. 



The Circular Pitch. 

The Circular Pit(^h. 

The Circular Pit(;h. 
The Circular Pitch. 
Thickness of Tooth. 



. Rule. 



Divide 3.1416 ))y the Diametral 
Pitch 

Divide Pitch Diameter by the 
product of .3183 and Number of 
Teeth 

Divide Outside Diameter by the 
product of .3183 and Number of 
Teeth plus 2 

The continued product of the 
Number of Teeth, the Circular 
Pitch and .3183 

Divide the product of Number of 
Teeth and Outside Diameter by 
Number of Teeth plus 2 . . . 

Subtract from the Outside Diame- 
ter the product of the Circular 
Pitch and .6366 

Multiply the Number of Teeth by 
the Addendum 

The continued product of the 
Number of Teeth plus 2, the 
Circular Pitch and .3183 . . . 

Add to the Pitch Diameter the 
product of the Circular Pitch 
and .6366 

Multiply Addendum by Number 
of Teeth plus 2 

Divide the product of Pitch Diam- 
eter and 3.1416 by the Circular 
Pitch ... 

One-half the Circular Pitch . . 

Multiply the Circular Pitch by 

.3183, or 8 = -5-' 

N 

INFultiply the Circular Pit(;h bv 
.3683 " 

Multiply the Circular Pitch b\ 
.6366 ■ 

Multiply the Circular Pitch by 
.6866 

iVfidtiply the Circular Pitch by .05 

One-tenth the Thickness of Tooth 
at Pitch Line 



Formula. 



P'= 



3.1416 
P 

D^ 
.3183 N 

D 

.3183 N+2 



D'=NP'.3183 



D'= 



ND 



N+2 
D'r=D— (P'.6366) 
D'=N8 

D=:(N+2)P'.3183 

D=D'-4-(P'.63G6) 




P' 



s = 


= P' 


3 183 


s + f = 


= P' .3683 


I)' 


= P 


' .()366 


D 


' + f 


= P'.6866 


f = 


t 


05 



f = 



177 



BROWN & SHARPE MFG. CO. 



TABLE OF TOOTH PARTS 



CIRCULAR PITCH IN FIRST COLUMN 



Jl 


Threads or 

Teeth per inch 

Linear. 


2 4 

1^ 


Thickness of 

Tooth on 
Pitch Line. 


^4 
II 


t 

r 


Depth of Space 

below 

Pitch Line. 


« 1 
> o 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


p' 


p' 


P 


t 


s 


D" 


s+f 


D"+/ 


P'X.3095 


P'X.3354 


2 


1 

2 


1.5708 


1.0000 


.6366 


1.2782 


.7366 


1.3732 


.6190 


.6707 


If 


8 
15 


1.6755 


.9375 


.5968 


1.1937 


.6906 


1.2874 


.5803 


.6288 


H 


4 

T 


1.7952 


.8750 


.5570 


1.1141 


.6445 


1.2016 


.5416 


.5869 


If 


8 


1.9333 


.8125 


.5173 


1.0345 


„5985 


L1158 


.5029 


.5450 


li 


f 


2.0944 


.7500 


.4775 


.9549 


.5525 


1.0299 


.4642 


.5030 


ih 


16 
-23 


2.1855 


.7187 


.4576 


.9151 


.5294 


.9870 


.4449 


.4821 


11 


-8. 
11 


2.2848 


.6875 


.4377 


.8754 


.5064 


.9441 


.4256 


.4611 


11 


A. 

4 


2.3562 


.6666 


.4244 


.8488 


.4910 


.9154 


.4127 


.4471 


1^ 


16 
21 


2.3936 


.6562 


.4178 


.8356 


.4834 


.9012 


.4062 


.4402 


H 


JL 
5 


2..5133 


.6250 


.3979 


.7958 


.4604 


.8583 


.3869 


.4192 


li 


16 
-19- 


2.6456 


.5937 


.3780 


.7560 


.4374 


.8154 


.3675 


.3982 


li 


8 
9 


2.7925 


.5625 


.3581 


.7162 


.4143 


.7724 


.3482 


.3773 


ifr 


16 

17 


2.9568 


.5312 


.3382 


.6764 


.3913 


.7295 


.3288 


-.3563 


1 


1 


3.1416 


.5000 


.3183 


.6366 


.3683 


.6866 


.3095 


.3354 


15 
16 


1* 


3.3510 


.4687 


.2984 


.5968 


.3453 


.6437 


.2902 


.3144 


7 
8 


li 


3.5904 


.4375 


.2785 


.5570 


.3223 


.6007 


.2708 


.2934 


# 


If 


3.8666 


.4062 


.2586 


.5173 


.2993 


.5579 


.2515 


.2725 


f 


11 


3.9270 


.4000 


.2546 


.5092 


._2946 


.5492 


.2476 


.2683 


3. 
4 


11 


4.1888 


.3750 


.2387 


.4775 


.2762 


.5150 


.2321 


.2515 


11 

• 16 


1^ 


4.5696 


.3437 


.2189 


.4377 


.2532 


.4720 


..2128 


.2306 


2 
3 


11 


4.7124 


.3333 


.2122 


.4244 


.2455 


.4577 


.2063 


.2236 


5 
8 


11 


5.0265 


.3125 


.1989 


.3979 


.2301 


.4291 


.1934 


.2096 


3 

5 


11 


5.2360 


.3000 


.1910 


.3820 


.2210 


.4120 


.1857 


.2012 


-4- 

7 


11 


5.4978 


.2857 


.1819 


.3638 


.2105 


.3923 


.1769 


.1916 


9 
16- 


11 


5.5851 


.2812 


.1790 


.3581 


.2071 


.3862 


.1741 


.1886 



To obtain the size of any part of a 
multiply the corresponding part of 1' 



circular pitch not given in the 
' pitch by the pitch required. 



table, 



178 



BROWN & SHARPE MFG. CO. 



TABLE OF TOOTH PARTS— Continued 



CIRCULAR PITCH IN FIRST COLUMN 





Threads or 

Teeth per inch 

Linear. 




TJiickness of 

Tooth on 
Pitch Line. 




t 

.a ^ 

r 


Depth of Space 

below 

Pitch Line. 




Width of 

Thread-Tool 

at End. 


o H 

SI 

^ s 


P' 


1" 
p' 


p 


t 


s 


D" 


s+f 


D'^/. 


P'X.30g5 


P'X.3354 


-|- 


2 


6.2832 


.2500 


.1592 


.3183 


.1842 


.3433 


.1547 


.1677 


~ 


21 


7.0685 


.2222 


.1415 


.2830 


.1637 


.3052 


.1376 


.1490 


IT 


2-f 


7.1808 


.2187 


.1393 


.2785 


.1611 


.3003 


.1354 


.1467 


-f- 


21- 


7.3304 


.2143 


.1364 


.2728 


.1578 


.2942 


.1326 


.1437 


2 

T 


21- 


7.8540 


.2000 


.1273 


.2546 


.1473 


.2746 


.1238 


.1341 


8 


2f 


8.3776 


.1875 


,1194 


.2387 


.1381 


.2575 


.1161 


.1258 


i 
11 


2f 


8.6394 


.1818 


.1158 


.2316 


.1340 


.2498 


.1125 


.1219 


1 


3 


9.4248 


.1666 


.1061 


.2122 


.1228 


.2289 


.1032 


.1118 


& 

16 


3i 


10.0531 


.1562 


.0995 


.1989 


.1151 


.2146 


.0967 


.1048 


8 
10 


3i 


10.4719 


.1500 


.0955 


.1910 


.1105 


.2060 


.0928 


.1006 


2 

7 


8i 


10.9956 


.1429 


.0909 


.1819 


.1052 


,1962 


.0884 


.0958 


1 

T 


4 


12.6664 


.1250 


.0796 


.1591 


.0921 


.1716 


.0774 


.0838 


9 

T 


4i 


14.1372 


.1111 


.0707 


.1415 


.0818 


.1526 


.0688 


.0745 


1 

5 


5 


15.7080 


.1000 


.0637 


.1273 


.0737 


.1373 


.0619 


.0671 


3 

10 


51- 


16.7552 


.0937 


.0597 


.1194 


.0690 


.1287 


.0580 


.0629 


11 


5f 


17.2788 


.0909 


.0579 


.1158 


.0670 


.1249 


.0563 


.0610 


1 

6 


6 


18.8496 


.0833 


.0531 


.1061 


.0614 


.1144 


.0516 


.0559 


2 

la 


6i 


20.4203 


.0769 


.0489 


.0978 


.0566 


.1055 


.0476 


.0516 


T 


7 


21.9911 


.0714 


.0455 


.0910 


.0526 


.0981 


.0442 


.0479 


2 
15 


7-1- 


23.5619 


.0666 


.0425 


.0850 


.0492 


.0917 


.0413 


.0447 


1 
T 


8 


25.1327 


.0625 


.0398 


.0796 


.0460 


.0858 


.0387 


.0419 


T 


9 


28.2743 


.0555 


.0354 


.0707 


.0409 


.0763 


.0344 


.0373 


JL 

10 


10 


31.4159 


.0500 


.0318 


.0637 


.0368 


.0687 


.0309 


.0335 


1 

16 


16 


50.2655 


.0312 


.0199 


.0398 


.0230 


.0429 


.0193 


.0210 


1 


20 


62.8318 


.0250 


.0159 


.0318 


.0184 


.0343 


.0155 


.0168 



To obtain the size of any part of a circular pitch 
multiply the corresponding part of 1" pitch by the 



not given in the table, 
pitch required. 



179 



BROWN & SHARPE MFG. CO. 



TABLE OF TOOTH PARTS 



DIAMETRAL PITCH IN FIRST COLUMN 



l-g 


1^ 
II 


Thickness 
of Tooth on 
Pitch Line. 


1^ 
<1 


t 

1 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


p 


P' 


t 


s 


D" 


s+f. 


D"+/. 


y2 


G.2832 


3.1416 


2.0000 


4.0000 


2.3142 


4.3142 


M 


4.1888 


2.0944 


1.3333 


2.6666 


1.5428 


2.8761 


1 


3.1416 


1.5708 


1.0000 


2.0000 


1 . 1571 


2.1571 


IM 


2.5133 


1.2566 


.8000 


1.6000 


.9257 


1.7257 


IK 


2.0944 


1.0472 


.6666 


1.3333 


.7714 


1.4381 


IH 


1.7952 


.8976 


.5714 


1.1429 


.6612 


1.2326 


2 


1.5708 


.7854 


.5000 


1.0000 


.5785 


1.0785 


2M 


1.3963 


.6981 


.4444 


.8888 


.5143 


.9587 


2K 


1.2566 


.6283 


.4000 


.8000 


.4628 


.8628 


2M 


1 . 1424 


.5712 


.3636 


.7273 


.4208 


.7844 


3 


1.0472 


.5236 


.3333 


.6666 


.3857 


.7190 


33^ 


.8976 


.4488 


.2857 


.5714 


.3306 


.6163 


4 


.7854 


.3927 


.2500 


.5000 


.2893 


.5393 


5 


.6283 


.3142 


.2000 


.4000 


.2314 


.4314 


6 


.5236 


.2618 


.1666 


.3333 


.1928 


.3595 


7 


.4488 


.2244 


.1429 


.2857 


.1653 


.3081 


8 


.3927 


.1963 


.1250 


.2500 


.1446 


.2696 


9 


.3491 


.1745 


.1111 


.2222 


.1286 


.2397 


10 


.3142 


.1571 


.1000 


.2000 


.1157 


.2157 


11 


.2856 


.1428 


.0909 


.1818 


.1052 


.1961 


12 


.2618 


.1309 


.0833 


.1666 


.0964 


.1798 


13 


.2417 


.1208 


.0769 


.1538 


.0890 


.1659 


14 


.2244 


.1122 


.0714 


.1429 


.0826 


.1541 



To obtain the size of any part of a diametral pitch not given in the table, 
divide the corresponding part of 1 diametral pitch by the pitch required. 

180 



BROWN & SHARPE MFG. CO. 



TABLE OF TOOTH P ART S^Continued 

DIAMETRAL PITCH IN FIRST COLUMN 





5^ 


Thickness 
of Tooth on 
Pitch Line. 


-^ or the 
Addendum 
or Module. 




Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


P. 


P'. 


t. 


s. 


D". 


s+f. 


D"./. 


15 


.2094 


.1047 


.0866 


.1333 


.0771 


.1438 


16 


.1963 


.0982 


.0625 


.1250 


.0723 


.1348 


17 


.1848 


.0924 


.0588 


.1176 


.0681 


.1269 


18 


.1745 


.0873 


.0555 


.1111 


.C643 


.1198 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1135 


20 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0455 


.0909 


.0526 


.0980 


24 


.1309 


.0654 


.0417 


.0833 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


.0561 


.0357 


.0714 


.0413 


.0770 


30 


.1047 


.0524 


.0333 


.0666 


.0386 


.0719 


32 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


34 


.0924 


.0462 


.0294 


.0588 


.0340 


.0634 


36 


.0873 


.0436 


.0278 


.0555 


.0321 


.0599 


38 


.0827 


.0413 


.0263 


.0526 


.0304 


.0568 


40 


.0785 


.0393 


.0250 


.0500 


.0289 


.0539 


42 


.0748 


.0374 


.0238 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 


46 


.0683 


.0341 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


56 


.0561 


.0280 


.0178 


.0357 


.0207 


.0385 


GO 


.0524 


.0262 


.0166 


.0333 


.0193 


.0360 



To obtain the size of any part 
divide the corresponding part of 



of a diametral pitch not given in the table, 
1 diametral pitch by the pitch required. 



181 



BROWN & SHARPE MFG. CO. 

TABLE FOR THE 
SOLUTION OF RIGHT ANGLED TRIANGLES 



SOLUTION OF TRIANGLES BY NATURAL LINES 



PARTS 


PARTS TO BE FOUND. 


GIVEN. 


Angle. 


Adj.- Side. 


Opp. Side. 


Hyp. 


Opp. Ang. 




^^-°^. 


, 








Opp. and 
Hyp. 


jHyp.2-0pp.2 


Co-il 




T-=^; 






, 




Opp. and 
Adj. 


J0pp.2+Adj.2 


Co'- = i^: 


Adj. and 
Hyp. 


/-, Adj . 

Cos. = .^ 

Hyp. 




J Hyp.2-Adj.2 




s'-ife 


Ang. and 
Opp. 




Opp.XCot. 




Opp. -^Sin. 


90°— Ang. 


Ang. and 
Adj. 






Adj. X Tang. 


Adj. -^ Cos. 


90°— Ang. 


Ang. and 
Hyp. 




Hyp. X Cos. 


Hyp. X Sin. 




90°— Ang. 





ADJ. 



OPP. 



ABBREVIATIONS USED 



Opp. = Opposite side. 
Adj. = Adjacent side. 
Hyp. = Hypothenuse. 
Ang. = Angle. 



Sin. = Sine. 

Tan. = Tangent. 

Cos. = Cosine. 

Cot. = Cotangent. 



182 



Natural Sines and Cosines 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



/ 








I 





2° 


3 


D 


4° 


r 

/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.00000 




.01745 


.99985 


.03490 


.99939 


.05234 


.99863 


.06976 


•99756 


60 


I 


,00029 






.01774 


.99984 


.03519 


.99938 


.05263 


.99861 


.07005 


.99754 


59 


2 


.00058 






.01803 


.99984 


.03548 


.99937 


.05292 


.99860 


.07034 


.99752 


58 


3 


.00087 






.01832 


.99983 


.03577 


.99936 


.05321 


.99858 


.07063 


.99750 


57 


4 


.00116 






.01862 


.99983 


.03606 


.99935 


•05350 


•99857 


.07092 


.99748 


S6 


5 


.00145 






.01891 


.99982 


.03635 


.99934 


.05379 


.99855 


.07121 


.99746 


5S 


6 


.00175 






.01920 


.99982 


.03664 


.99933 


.05408 


.99854 


.07150 


.99744 


54 


7 


.00204 






.01949 


.99981 


.03693 


.99932 


.05437 


.99852 


.07179 


.99742 


S3 


8 


.00233 






.01978 


.99980 


.03723 


.99931 


.05466 


.99851 


.07208 


.99740 


52 


9 


.00262 






.02007 


.99980 


.03752 


.99930 


.05495 


.99849 


.07237 


.99738 


51 


10 


.00291 






.02036 


.99979 


.03781 


.99929 


.05524 


.99847 


.07266 


.997.36 


50 


II 


.00320 


.99999 


.02065 


.99979 


.03810 


.99927 


.05553 


.99846 


.07295 


.99734 


49 


12 


.00349 


.99999 


.02094 


.99978 


.03839 


.99926 


.05582 


.99844 


.07324 


.99731 


48 


13 


.00378 


.99999 


.02123 


.99977 


.03868 


.99925 


.05611 


.99842 


.07353 


•99729 


47 


14 


.00407 


.99999 


.02152 


.99977 


.03897 


.99924 


.05640 


.99841 


.07382 


•99727 


46 


IS 


.00436 


.99999 


.02181 


.99976 


.03926 


.99923 


.05669 


.99839 


.07411 


.99725 


45 


i6 


.00465 


.99999 


.02211 


.99976 


.03955 


.99922 


.05698 


.99838 


.07440 


.99723 


44 


17 


.00495 


.99999 


.02240 


.99975 


.03984 


.99921 


.05727 


.99836 


.07469 


.99721 


43 


i8 


.00524 


.99999 


.02269 


.99974 


.04013 


.99919 


.05756 


.99834 


.07498 


.99719 


42 


19 


.00553 


.99998 


.02298 


.99974 


,.04042 


.99918 


.05785 


.99833 


.07527 


.99716 


41 


20 


.00582 


.99998 


.02327 


.99973 


.04071 


.99917 


.05814 


.99831 


.07556 


.99714 


40 


21 


.00611 


.99998 


.02356 


.99972 


.04100 


.99916 


.05844 


.99829 


.07585 


.99712 


39 


22 


.00640 


.99998 


.02385 


.99972 


.04129 


.99915 


.05873 


.99827 


.07614 


.99710 


38 


23 


.00669 


.99998 


.02414 


.99971 


.04159 


.99913 


.05902 


.99826 


•07643 


.99708 


37 


24 


.00698 


.99998 


.02443 


.99970 


.04188 


.99912 


.05931 


.99824 


.07672 


.99705 


36 


25 


.00727 


.99997 


.02472 


.99969 


.04217 


.99911 


.05960 


.99822 


.07701 


.99703 


3S 


26 


.00756 


.99997 


.02501 


.99969 


.04246 


.99910 


.05989 


.99821 


.07730 


.99701 


34 


27 


.00785 


.99997 


.02530 


.99968 


.04275 


.99909 


.06018 


.99819 


.07759 


.99699 


33 


28 


.00814 


.99997 


.02560 


.99967 


.04304 


.99907 


.06047 


•99817 


.07788 


.99696 


32 


29 


.00844 


.99996 


.02589 


.99966 


.04333 


.99906 


.06076 


.998x5 


.07817 


.99694 


31 


30 


.00873 


.99996 


.02618 


.99966 


.04362 


.99905 


.06105 


.99813 


.07846 


.99692 


30 


31 


.00902 


.99996 


.02647 


.99965 


.04391 


.99904 


.06134 


.99812 


.07875 


.99689 


29 


32 


.00931 


.99996 


.02676 


.99964 


.04420 


.99902 


.06163 


.99810 


.07904 


.99687 


28 


33 


.009O0 


.99995 


.02705 


.99963 


.04449 


.99901 


.06192 


.99808 


.07933 


.99685 


27 


34 


.00989 


.99995 


.02734 


.99963 


.04478 


.99900 


.06221 


.99806 


.07962 


•99683 


26 


35 


.01018 


.99995 


.02763 


.99962 


.04507 


.99898 


.06250 


.99804 


.07991 


.99680 


'25 


36 


.01047 


.99995 


.02792 


.99961 


.04536 


.99897 


.06279 


.99803 


.08020 


.99678 


24 


37 


.01076 


.99994 


.02821 


.99960 


.04565 


.99896 


.06308 


.99801 


.08049 


.99676 


2Z 


38 


.01105 


.99994 


.02850 


.99959 


.04594 


.99894 


.06337 


.99799 


.08078 


.99673 


22 


39 


.01134 


.99994 


.02879 


.99959 


.04623 


.99893 


.06366 


.99797 


.08107 


.99671 


21 


40 


.01164 


.99993 


.02908 


.99958 


.04653 


.99892 


.06395 


•99795 


.08136 


.99668 


20 


41 


.01193 


.99993 


.02938 


.99957 


.04682 


.99890 


.06424 


.99793 


.08165 


.99666 


19 


42 


.01222 


.99993 


.02967 


.99956 


.04711 


.99889 


•06453 


.99792 


.08194 


.99664 


18 


43 


.01251 


.99992 


.02996 


.99955 


.04740 


.99888 


.06482 


.99790 


.08223 


.99661 


17 


44 


.01280 


.99992 


.03025 


.99954 


.04769 


.99886 


.06511 


.99788 


.08252 


.99659 


16 


45 


.01309 


•99991 


.03054 


.99953 


.04798 


.99885 


.06540 


.99786 


.08281 


.99657 


IS 


46 


.01338 


.99991 


.03083 


.99952 


.04827 


.99883 


•06569 


.99784 


.08310 


.99654 


14 


47 


.01367 


.99991 


.03112 


.99952 


.04856 


.99882 


.06598 


.99782 


.08339 


.99652 


13 


48 


.01396 


.99990 


.03141 


.99951 


.04885 


.99881 


.06627 


.99780 


.08368 


.99649 


12 


49 


.01425 


.99990 


.03170 


.99950 


.04914 


.99879 


.06656 


.99778 


.08397 


.99647 


II 


50 


.01454 


.99989 


.03199 


.99949 


.04943 


.99878 


.06685 


•99776 


.08426 


.99644 


10 


51 


.01483 


.99989 


.03228 


.99948 


.04972 


.99876 


.06714 


.99774 


.08455 


.99642 


9 


52 


.01513 


.99989 


.03257 


.99947 


.05001 


.99875 


•06743 


.99772 


.08484 


.99639 


8 


53 


.01542 


.99988 


.03286 


.99946 


.05030 


.99873 


•06773 


.99770 


.08513 


.99637 


7 


54 


.01571 


.99988 


.03316 


.99945 


.05059 


.99872 


.06802 


.99768 


.08542 


.9963s 


6 




.01600 


.99987 


.03345 


.99944 


.05088 


.99870 


.06831 


.99766 


.08571 


.99632 


S 


56 


.01629 


.99987 


.03374 


.99943 


.05117 


.99869 


.06860 


•99764 


.08600 


.99630 


4 


n 


.01658 


.99986 


.03403 


.99942 


.05146 


.99867 


.06889 


.99762 


.08629 


•99627 


3 


58 


.01687 


.99986 


.03432 


.99941 


.05175 


.99866 


.06918 


.99760 


.08658 


.99625 


2 


59 


.01716 


.99985 


.03461 


.99940 


.05205 


.99864 


.06947 


•99758 


.08687 


.99622 


I 


60 


.01745 


.99985 


.03490 


.99939 


.05234 


.99863 


.06976 


.99756 


.08716 


.99619 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


8c 


)° 


8^ 


5° 


87 





86 





8^ 


-0 



184 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



1 


5 





6 





7 





8° 


9° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


Cosine 


Sine C 


'osine 





.08716 


.99619 


.10453 


•99452 


.12187 


.59255 


• 13917 


99027 


.15643 


98769 


60 


1 


.08745 


.99617 


.10482 


.99449 


.12216 


•99251 


.13946 


99023 


.15672 


98764 


59 


2 


.08774 


.99614 


.10511 


•99446 


.12245 


.99248 


.13975 


99019 


.15701 


98760 


58 


3 


.08803 


.99612 


.10540 


.99443 


.12274 


•99244 


.14004 


99015 


.15730 


98755 


57 


4 


.08831 


.99609 


.10569 


.99440 


.12302 


.99240 


.14033 


99011 


.15758 


98751 


56 




.08860 


.99607 


.10597 


•99437 


• 12331 


.99237 


.14061 


99006 


.15787 


98746 


55 


6 


.08889 


.99604 


.10626 


•99434 


.12360 


.99233 


.14090 


99002 


.15816 


9874 X 


54 


7 


.08918 


.99602 


• 10655 


•99431 


.12389 


.99230 


.14119 


98998 


.15845 


98737 


53 


8 


.08947 


.99599 


.10684 


• 99428 


.12418 


.99226 


.14148 


98994 


.15873 


98732 


52 


9 


.08976 


.99596 


.10713 


•99424 


.12447 


.99222 


.X4177 


98990 


.15902 


98728 


SX 


10 


.09005 


.99594 


.10742 


.99421 


.12476 


.99219 


.14205 


98986 


.15931 


98723 


50 


II 


.09034 


.99591 


.10771 


.99418 


.12504 


.992x5 


.X4234 


98982 


.15959 


98718 


49 


12 


.09063 


.99588 


.10800 


•99415 


.12533 


.9921 X 


.14263 


98978 


.15988 


98714 


48 


13 


.09092 


.99586 


.10829 


.99412 


.12562 


.99208 


.14292 


98973 


.16017 


98709 


47 


14 


.09121 


•99583 


.10858 


•99409 


.12591 


.99204 


.14320 


98969 


.16046 


98704 


46 


IS 


.09150 


.99580 


.10887 


.99406 


.12620 


.99200 


.14349 


9896s 


.16074 


98700 


45 


i6 


.09179 


.99578 


.10916 


.99402 


.12649 


.99197 


.14378 


98961 


.16103 


98695 


44 


17 


.09208 


.99575 


.10945 


.99399 


.12678 


•99193 


.14407 


98957 


.16132 


98690 


43 


i8 


.09237 


.99572 


•10973 


.99396 


.12706 


.99189 


.14436 


98953 


.16160 


98686 


42 


19 


.09266 


.99570 


.11002 


■ 99393 


• 12735 


.99186 


.14464 


98948 


.16189 


98681 


41 


20 


.09295 


.99567 


.11031 


•99390 


.12764 


.99182 


.14493 


98944 


.16218 


9B676 


40 


21 


.09324 


.99564 


.11060 


.99386 


• 12793 


.99178 


.14522 


98940 


.16246 


98671 


39 


22 


.09353 


.99562 


.11089 


.99383 


.12822 


.99175 


.14551 


98936 


.16275 


98667 


38 


23 


.09382 


.99559 


.11118 


.99380 


.12851 


.99171 


.14580 


98931 


.16304 


98662 


37 


24 


.09411 


.99556 


.11147 


• 99377 


.12880 


.99167 


.14608 


98927 


.16333 


98657 


36 


25 


.09440 


.99553 


.11176 


•99374 


.12908 


.99163 


.14637 


98923 


.16361 


98652 


35 


26 


.09469 


.99551 


• 11205 


•99370 


.12937 


.99160 


.14666 


98919 


.16390 


98648 


34 


27 


.09498 


.99548 


.11234 


•99367 


.12966 


.99156 


.14695 


98914 


.16419 


98643 


33 


28 


.09527 


.99545 


.11263 


•99364 


.12995 


.99152 


.14723 


98910 


.16447 


98638 


32 


29 


.09556 


.99542 


.11291 


•99360 


.13024 


.99148 


.14752 


98906 


.16476 


98633 


31 


30 


.09585 


.99540 


.11320 


•99357 


•13053 


•99144 


.14781 


98902 


.16505 


98629 


30 


31 


.09614 


.99537 


.11349 


.99354 


.13081 


.99I4X 


.X4810 


98897 


.16533 


98624 


29 


32 


.09642 


•99534 


• 11378 


.99351 


.13110 


.99137 


.14838 


98893 


.16562 


98619 


28 


33 


.09671 


•99531 


.11407 


.99347 


• 13139 


.99133 


.14867 


98889 


.16591 


98614 


27 


34 


.09700 


.99528 


.11436 


.99344 


.13168 


.99x29 


.14896 


98884 


.16620 


98609 


26 


35 


.09729 


.99526 


.11465 


.99341 


•13197 


.99125 


.14925 


98880 


.16648 


98604 


25 


36 


.09758 


•99523 


•I1494 


.99337 


.13226 


.99122 


.14954 


98876 


.16677 


98600 


24 


37 


.09787 


• 99520 


■ 11523 


•99334 


• 13254 


.99118 


.14982 


98871 


.16706 


98595 


23 


38 


.09816 


.99517 


•IISS2 


•99331 


•13283 


.99114 


.15011 


98867 


.16734 


98590 


22 


39 


.09845 


.99514 


.11580 


•99327 


• 13312 


.99110 


.15040 


98863 


.16763 


98585 


21 


40 


.09874 


.99511 


.11609 


.99324 


• 13341 


.99106 


.15069 


98858 


.16792 


98580 


20 


41 


.09903 


.99508 


.11638 


.99320 


•13370 


.99102 


.15097 


98854 


.16820 


9857s 


19 


42 


.09932 


•99506 


.11667 


•99317 


.13399 


.99098 


.15126 


98849 


.16849 


98570 


18 


43 


.09961 


.99503 


.11696 


.99314 


• 13427 


.99094 


• X515S 


98845 


.16878 


98565 


17 


44 


.09990 


.99500 


.11725 


.99310 


• 13456 


•99091 


.X5184 


98841 


.16906 


98561 


16 


45 


.10019 


.99497 


•II754 


.99307 


'3485 


• 99087 


.15212 


98836 


.16935 


98556 


IS 


46 


.10048 


.99494 


.11783 


•99303 


• 13514 


•99083 


.15241 


98832 


.16964 


98551 


14 


47 


.10077 


.99491 


.11812 


.99300 


•13543 


.99079 


.15270 


98827 


.16992 


98546 


13 


48 


.10106 


.99488 


.11840 


•99297 


.13572 


•99075 


.15299 


98823 


.17021 


98541 


12 


49 


.10X35 


.99485 


.11869 


.99293 


.13600 


•99071 


• 15327 


98818 


.17050 


98536 


II 


SO 


.10164 


.99482 


.11898 


.99290 


.13629 


.99067 


• 15356 


98814 


.17078 


98531 


10 


51 


.10192 


.99479 


.11927 


.99286 


.13658 


•99063 


• 15385 


98809 


.17x07 


98526 


9 


S2 


.10221 


.99476 


.11956 


.99283 


.13687 


.99059 


.15414 


98805 


.17136 


98521 


8 


S3 


.10250 


.99473 


.11985 


.99279 


•13716 


.99055 


.15442 


98800 


.17164 


98516 


7 


S4 


.10279 


.99470 


.12014 


.99276 


• 13744 


.99051 


.15471 


98796 


.17193 


98511 


6 


SS 


.10308 


.99467 


.12043 


.99272 


•13773 


•99047 


.15500 


98791 


.17222 


98506 


5 


56 


.10337 


.99464 


.12071 


.99269 


.13802 


•99043 


.15529 


98787 


.17250 


98501 


4 


57 


.10366 


.99461 


.12100 


•99265 


.13831 


.99039 


.15557 


98782 


.17279 


98496 


3 


58 


.10395 


•99458 


.12129 


.99262 


.13860 


•99035 


.15586 


98778 


•17308 


98491 


2 


59 


.10424 


•99455 


.12.58 


.99258 


.13889 


.99031 


• IS615 


98773 


.17336 


98486 


I 


60 


.10453 


.99452 


.12187 


.99255 


.13917 


■99027 


• 15643 


98769 


.17365 


98481 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


8. 


^° 


8: 


5° 


8i 


2° 


81^ 


3 


80^ 


> 



185 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



/ 


10 





II 





12 





13 





M" 1 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


•17365 


.98481 


.19081 


.98163 


.20791 


.97815 


.22495 


•97437 


.24192 


.97030 


60 


I 


.17393 


.98476 


.19109 


.98157 


.20820 


.97809 


.22523 


.97430 


.24220 


.97023 


59 


2 


.17422 


.98471 


.19138 


.98152 


.20848 


.97803 


.22552 


.97424 


.24249 


•97015 


58 


3 


.I74SI 


.98466 


.19167 


.98146 


.20877 


.97797 


.22580 


.97417 


•24277 


.97008 


57 


4 


.17479 


.98461 


.19195 


.98140 


.20905 


.97791 


.22608 


•974 1 1 


.24305 


.97001 


S6 


S 


.17508 


.98455 


.19224 


.98135 


.20933 


.97784 


.22637 


•97404 


.24333 


.96994 


55 


6 


.17537 


.98450 


.19252 


.98129 


.20962 


.97778 


.22665 


.97398 


.24362 


.96987 


54 


7 


.17565 


.98445 


.19281 


.98124 


.20990 


.97772 


.22693 


.97391 


.24390 


.96980 


53 


8 


.17594 


.98440 


•19309 


.98118 


.21019 


.97766 


.22722 


.97384 


.24418 


.96973 


52 


9 


.17623 


.98435 


.19338 


.98112 


.21047 


.97760 


.22750 


.97378 


.24446 


.96966 


51 


10 


.17651 


.98430 


.19366 


.98107 


.21076 


.97754 


.22778 


.97371 


.24474 


.96959 


SO 


II 


.17680 


.98425 


.19395 


.98101 


.21104 


.97748 


.22807 


.97365 


.24503 


.96952 


49 


12 


.17708 


.98420 


•19423 


.98096 


.21132 


.97742 


.22835 


.97358 


.24531 


.96945 


48 


13 


.17737 


.98414 


•19452 


.98090 


.21161 


.97735 


.22863 


.97351 


.24559 


.96937 


47 


14 


.17766 


.98409 


.19481 


.98084 


.21189 


.97729 


.22892 


•97345 


.24587 


.96930 


46 


IS 


• 17794 


.98404 


.19509 


.98079 


.21218 


•97723 


.22920 


•97338 


.24615 


.96923 


45 


i6 


.17823 


.98399 


.19538 


.98073 


.21246 


•97717 


.22948 


•97331 


.24644 


.96916 


44 


17 


.17852 


.98394 


.19566 


.98067 


.21275 


.97711 


.22977 


•97325 


.24672 


.96909 


43 


i8 


.17880 


.98389 


•I9S9S 


.98061 


.21303 


.97705 


.2300s 


•97318 


.24700 


.96902 


42 


19 


.17909 


.98383 


.19623 


.98056 


.21331 


.97698 


.23033 


.97311 


.24728 


.96894 


41 


20 


• 17937 


.98378 


.19652 


.98050 


.21360 


.97692 


.23062 


.97304 


•24756 


.96887 


40 


21 


.17966 


.98373 


.19680 


.98044 


.21388 


.97686 


.23090 


.97298 


.24784 


.96880 


39 


22 


.17995 


.98368 


.19709 


.98039 


.21417 


.97680 


.23118 


.97291 


.24813 


.96873 


38 


23 


.18023 


.98362 


.19737 


.98033 


•21445 


.97673 


.23146 


.97284 


.24841 


.96866 


37 


24 


.18052 


.98357 


.19766 


.98027 


.21474 


.97667 


.23175 


.97278 


.24869 


.96858 


36 


25 


.18081 


.98352 


.19794 


.98021 


.21502 


.97661 


.23203 


•97271 


.24897 


.96851 


35 


26 


.i8iog 


.98347 


.19823 


.98016 


.21530 


•97655 


.23231 


•97264 


.24925 


.96844 


34 


21 


.18138 


.98341 


.19851 


.98010 


.21559 


.97648 


.23260 


•97257 


.24954 


.96837 


33 


28 


.18166 


.98336 


.19880 


.98004 


.21587 


.97642 


.23288 


.97251 


.24982 


.96829 


32 


29 


.18195 


.98331 


.19908 


.97998 


.21616 


.97636 


.23316 


.97244 


.25010 


.96822 


31 


30 


.18224 


.98325 


.19937 


.97992 


.21644 


.97630 


.23345 


.97237 


.25038 


.96815 


30 


31 


.18252 


.98320 


• 1996s 


•97987 


.21672 


•97623 


.23373 


.97230 


.25066 


.96807 


^l 


32 


.18281 


.983 IS 


.19994 


.97981 


.21701 


•97617 


.23401 


.97223 


.25094 


.96800 


28 


33 


.18309 


.98310 


.20022 


.97975 


.21729 


.97611 


.23429 


.97217 


.25122 


.96793 


21 


34 


.18338 


.98304 


.20051 


.97969 


.21758 


•97604 


.23458 


.97210 


.25151 


.96786 


26 


35 


.18367 


.98299 


.20079 


.97963 


.21786 


•97598 


.23486 


.97203 


.25179 


.96778 


25 


36 


.18395 


•98294 


.20108 


.97958 


.21814 


•97592 


.23514 


.97196 


.25207 


.96771 


24 




.18424 




.20136 


.97952 


.21843 


.97585 


.23542 


.97189 


.25235 


.96764 


23 


38 


.18452 


^98283 


.20165 


.97946 


.21871 


.97579 


.23571 


.97182 


.25263 


.96756 


22 


39 


.18481 


.98277 


.20193 


.97940 


.21899 


.97573 


.23599 


.97176 


.25291 


.96749 


21 


40 


.18509 


.98272 


.20222 


.97934 


.21928 


.97566 


.23627 


.97169 


.25320 


.96742 


20 


41 


.18538 


.98267 


.20250 


.97928 


.21956 


.97560 


.23656 


.97162 


.25348 


.96734 


19 


42 


.18567 


.98261 


.20279 


.97922 


.21985 


.97553 


.23684 


.97155 


.25376 


.96727 


18 


43 


.18595 


.98256 


.20307 


.97916 


.22013 


.97547 


.23712 


.97148 


.25404 


.96719 


^l 


44 


.18624 


.98250 


.20336 


.97910 


.22041 


•97541 


.23740 


.97141 


.25432 


.96712 


16 


45 


.18652 


.98245 


.20364 


•9790s 


.22070 


.97534 


.23769 


.97134 


.25460 


.9670s 


IS 


46 


.18681 


.98240 


.20393 


•97899 


.22098 


.97528 


.23797 


.97127 


.25488 


.96697 


14 


47 


.18710 


.98234 


.20421 




.22126 


.97521 


.23825 


.97120 


.25516 


.96690 


13 


48 


.18738 


.98229 


.20450 


•97887 


.22155 


.97515 


.23853 


.97113 


.25545 


.96682 


12 


49 


.18767 


.98223 


.20478 


.97881 


.^2183 


.97508 


.23882 


.97106 


.25573 


.96675 


11 


SO 


.18795 


.98218 


.20507 


.97875 


.22212 


.97502 


.23910 


.97100 


.25601 


.96667 


10 


51 


.18824 


.98212 


.20535 


.97869 


.22240 


.97496 


.23938 


.97093 


.25629 


.96660 


9 


S2 


.18852 


.98207 


.20563 


.97863 


.22268 


.97489 


.23966 


.97086 


.25657 


.96653 


8 


S3 


.18881 


.98201 


.20592 


.97857 


.22297 


.97483 


.23995 


.97079 


.2568s 


.96645 


7 


54 


.18910 


.98196 


.20620 


.97851 


.22325 


.97476 


.24023 


.97072 


.25713 


.96638 


6 


55 


.18938 


.98190 


.20649 


.97845 


.22353 


.97470 


.24051 


.97065 


.25741 


.96630 


5 


56 


.18967 


.98185 


.20677 


.97839 


.22382 


.97463 


•24079 


.97058 


.25769 


.96623 


4 


57 


.18995 


.98179 


.20706 


.97833 


.22410 


.97457 


.24108 


.97051 


.25798 


.96615 


3 


S8 


.19024 


.98174 


.•20734 


.97827 


.22438 


.97450 


.24136 


.97044 


.25826 


.96608 


2 


59 


.19052 


.98168 


.20763 


.97821 


.22467 


.97444 


.24164 


•97037 


^5854 


.96600 


I 


60 


.19081 


.98163 


.20791 


.97815 


.22495 


.97437 


.24192 


•97030 


.25882 


.96593 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


f 


7< 


t 


7^ 


^° 


7 


f 


7< 


3 


7 


5° 



186 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



/ 


15° 


16° 


17° 


i8<^ 


19° 


1 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.25882 


.96593 


.27564 


.96126 


.29237 


.95630 


.30902 


.95106 


.32557 


.94552 


60 


I 


.25910 


.96585 


.27592 


.96118 


.29265 


.95622 


.30929 


.95097 


.32584 


.94542 


59 


2 


.25938 


.96578 


.27620 


.96110 


.29293 


.95613 


.30957 


.95088 


.32612 


.94533 


58 


3 


.25966 


.96570 


.27648 


.96102 


.29321 


.95605 


.30985 


.95079 


.32639 


.94523 


57 


4 


.25994 


.96562 


.27676 


.96094 


.29348 


.95596 


.31012 


.95070 


.32667 


.94514 


56 


S 


.26022 


.96555 


.27704 


.96086 


.29376 


.95588 


.31040 


.95061 


.32694 


.94504 


55 


6 


.26050 


.96547 


.27731 


.96078 


.29404 


.95579 


.31068 


.95052 


.32722 


.94495 


54 


7 


.26079 


.96540 


.27759 


.96070 


.29432 


.95571 


.31095 


.95043 


.32749 


.94485 


53 


8 


.26107 


.96532 


.27787 


.96062 


.29460 


.95562 


.31123 


.95033 


.32777 


.94476 


52 


9 


.26135 


.96524 


.27815 


.96054 


.29487 


.95554 


.31151 


.95024 


.32804 


.94466 


51 


10 


.26163 


.96517 


.27843 


.96046 


•29515 


.95545 


.31178 


.95015 


.32832 


•94457 


50 


II 


.26191 


.96509 


.27871 


.96037 


.29543 


.95536 


.31206 


.95006 


.32859 


.94447 


49 


12 


.26219 


.96502 


.27899 


.96029 


.29571 


.95528 


.31233 


.94997 


.32887 


.94438 


48 


13 


.26247 


.96494 


.27927 


.96021 


.29599 


.95519 


.31261 


.94988 


.32914 


.94428 


47 


14 


.26275 


.96486 


.27955 


.96013 


.29626 


.95511 


.31289 


.94979 


.32942 


.94418 


46 


IS 


.26303 


.96479 


.27983 


.96005 


.29654 


.95502 


.31316 


.94970 


.32969 


.94409 


45 


i6 


.26331 


.96471 


.28011 


.95997 


.29682 


.95493 


.31344 


.94961 


.32997 


.94399 


44 


17 


.26359 


.96463 


.28039 


.95989 


.29710 


.95485 


.31372 


■94952 


.33024 


.94390 


43 


i8 


.26387 


.96456 


.28067 


.95981 


.29737 


.95476 


.31399 


.94943 


.33051 


.94380 


42 


19 


.26415 


.96448 


.28095 


.95972 


.29765 


.95467 


.31427 


.94933 


.33079 


.94370 


41 


20 


.26443 


.96440 


.28123 


.95964 


.29793 


.95459 


.31454 


.94924 


.33106 


.94361 


40 


21 


.26471 


.96433 


.28150 


.95956 


.29821 


.95450 


.31482 


.9491S 


.33134 


.94351 


39 


22 


.26500 


.96425 


.28178 


.95948 


.29849 


.95441 


.31510 


.94906 


.33161 


.94342 


38 


23 


.26528 


.96417 


.28206 


.95940 


.29876 


.95433 


.31537 


.94897 


.33189 


.94332 


il 


24 


.26556 


.96410 


.28234 


.95931 


.29904 


.95424 


.31565 


.94888 


.33216 


.94322 


36 


25 


.26584 


.96402 


.28262 


.95923 


.29932 


.95415 


.31593 


.94878 


.33244 


.94313 


35 


26 


.26612 


.96394 


.28290 


.95915 


.29960 


.95407 


.31620 


.94869 


.33271 


.94303 


34 


27 


.26640 


.96386 


.28318 


.95907 


.29987 




.31648 


.94860 


.33298 


.94293 


33 


28 


.26668 


.06379 


.28346 


.95898 


.30015 


.95389 


.31675 


.94851 


.33326 


.94284 


32 


29 


.26696 


.96371 


.28374 


.95890 


.30043 


.95380 


.31703 


.94842 


.33353 


.94274 


31 


30 


.26724 


.96363 


.28402 


.95882 


.30071 


.95372 


.31730 


.94832 


.33381 


.94264 


30 


31 


.26752 


.96355 


.28429 


.95874 


.30098 


.95363 


.31758 


.94823 


.33408 


.94254 


29 


32 


.26780 


.96347 


.28457 


.95865 


.30126 


.95354 


.31786 


.94814 


.33436 


.94245 


28 


33 


.26808 


.96340 


.28485 


.95857 


.30154 


.95345 


.31813 


.94805 


.33463 


.94235 


27 


34 


.26836 


.96332 


.28513 


.95849 


.30182 


.35537 


.31841 


.94795 


.33490 


.94225 


26 


35 


.26864 


.96324 


.^8s4i 


.95841 


.30209 


.95328 


.31868 


.94786 


.33518 


.94215 


25 


36 


.26892 


.96316 


.28569 


.95832 


.30237 


.95319 


.3189:: 


.94777 


.33545 


.94206 


24 


37 


.26920 


.96308 


.28597 


.95824 


.30265 


.95310 


.31923 


.947C8 


.33573 


.94196 


23 


38 


.26948 


.96301 


.28625 


.95816 


.30292 


.95301 


.31951 


.94753 


.33600 


.94186 


22 


39 


.26976 


.96293 


.28652 


.95807 


.30320 


.95293 


.31979 


.94749 


.33627 


.94176 


21 


40 


.27004 


.96285 


.28680 


.95799 


.30348 


.95284 


.32006 


.94740 


.33655 


.94167 


20 


41 


.27032 


.96277 


.28708 


.95791 


.30376 


.95275 


.32034 


.94730 


.33682 


•94157 


19 


42 


.27060 


.96269 


.28736 


.95782 


.30403 


.95266 


.32061 


.94721 


.33710 


.94147 


18 


43 


.27088 


.96261 


.28764 


..95774 


.30431 


.95257 


.32089 


.94712 


.33737 


.94137 


17 


44 


.27116 


.96253 


.28792 


.95766 


.30459 


.95248 


.32116 


.94702 


.33764 


.94127 


16 


45 


.27144 


.96246 


.28820 


.95757 


.30486 


.95240 


.32144 


.94693 


.33792 


.94118 


15 


46 


.27172 


.96238 


.28847 


.95749 


.30514 


.95231 


.32171 


.94684 


.33819 


.94108 


14 


^Z 


.27200 


.96230 


.28875 


.95740 


.30542 


.95222 


.32199 


.94674 


■33846 


.94098 


13 


48 


.27228 


.96222 


.28903 


.95732 


.30570 


.95213 


.32227 


.94665 


.33874 


.94088 


12 


49 


.27256 


.96214 


.28931 


.95724 


.30597 


.95204 


.32254 


.94656 


.33901 


.94078 


II 


SO 


.27284 


.96206 


.28959 


.95715 


.30625 


.95195 


.,32282 


.94646 


.33929 


.94068 


10 


SI 


.27312 


.96198 


.28987 


.95707 


.30653 


.95186 


.32309 


■94637 


.33956 


.94058 


9 


52 


.27340 


.96190 


.29015 


.95698 


.30680 


.95177 


.32337 


.94627 


.33983 


.94049 


8 


S3 


.27368 


.96182 


.29042 


.•95690 


.30708 


.95168 


.32364 


.94618 


.34011 


.94039 


7 


54 


.27396 


.96174 


.29070 


.95681 


.30736 


■95159 


.32392 


.94609 


.34038 


.94029 


6 


55 


.27424 


.96166 


.29098 


.95673 


.30763 


.95150 


.32419 


.94599 


.34065 


.94019 


5 


56 


.27452 


.96158 


.29126 


.95664 


.30791 


.95142 


.32447 


.94590 


.34093 


.94009 


4 


H 


.27480 


.96150 


.29154 


.95656 


.30819 


.95133 


.32474 


.94580 


.34120 


.93999 


3 


S8 


.27508 


.96142 


.29182 


.95647 


.30846 


.95124 


.32502 


.94571 


.34147 


.93989 


2 


59 


.27536 


•96i3< 


' .29209 


.95639 


.30874 


.95115 


.32529 


.94561 


•34175 


.93979 


I 


60 


.27564 


.96126 


.29237 


.95630 


.30902 


.95106 


.32557 


.94552 


.34202 


.93969 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


7' 


4° 


7 


5° 


7' 


2° 


7 


1° 


7 


0° 



187 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



■ 


20° 


21 





22^ 


23° 


24° 


1 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


.34202 


.93969 


.35837 


.93358 


.37461 


.92718 


.39073 


.920C0 


.40674 


•91355 


60 


I 


.34229 


.93959 


.35864 


.93348 


.37488 


.92707 


.39100 


.92039 


.40700 


.91343 


59 


2 


.34257 


.93949 


.35891 


.93337 


.37515 


.92697 


.39127 


.92028 


.40727 


.91331 


58 


3 


.34284 


•93939 


.35918 


■93327 


.37542 


.92686 


.39153 


.92016 


.40753 


.91319 


57 


4 


.34311 


•93929 


.35945 


.93316 


.37569 


.92675 


.39180 


.92005 


.40780 


.91307 


56 


S 


.34339 


•93919 


.35973 


.93306 


.37595 


.92664 


•39207 


.91994 


.40806 


.91295 


55 


6 


.34366 


.93909 


.36000 


.93295 


.37622 


•92653 


.39234 


.91982 


.40833 


.91283 


54 


7 


.34393 


•93899 


.36027 


.93285 


.37649 


.92642 


.39260 


.91971 


.40860 


.91272 


53 


8 


.34421 


.93889 


.36054 


.93274 


.37676 


•92631 


.39287 


.91959 


.40886 


.91260 


52 


9 


.34448 


.93879 


.36081 


.93264 


.37703 


.92620 


.39314 


•91948 


.40913 


.91248 


51 


lO 


.34475 


.93869 


.36108 


.93253 


.37730 


.92609 


.39341 


.91936 


.40939 


.91236 


50 


II 


.34503 


.93859 


.36135 


.93243 


.37757 


.92598 


•39367 


.91925 


.40966 


.91224 


49 


12 


.34530 


.93849 


.36162 


.93232 


.37784 


•92587 


.39394 


.91914 


.40992 


.91212 


48 


13 


.34557 


.93839 


.36190 


.93222 


.37811 


.92576 


.39421 


.91902 


.41019 


.91200 


47 


14 


.34584 


.93829 


.36217 


.93211 


•37838 


.92565 


.39448 


.91891 


.41045 


.91188 


46 


IS 


.34612 


.93819 


.36244 


.93201 


•37865 


.92554 


.39474 


.91879 


.41072 


.91176 


45 


i6 


.34639 


•93809 


.36271 


.93 J 90 


•37892 


.92543 


.3950X 


.91868 


.41098 


.91164 


44 


17 


.34666 


.93799 


.36298 


.93180 


.37919 


.92532 


.39528 


.91856 


.41125 


.91152 


43 


i8 


.34694 


.93789 


.36325 


.93169 


.37946 


•92521 


.39555 


.91845 


.41151 


.91140 


42 


19 


• 34721 


.93779 


.36352 


.93159 


.37973 


•92510 


.39581 


.91833 


.41178 


.91128 


41 


20 


.34748 


.93769 


.363'79 


.93148 


.37999 


.92499 


.39608 


.91822 


.41204 


.91116 


40 


21 


.34775 


•93759 


.36406 


.93137 


.38026 


.92488 


.39635 


.91810 


.41231 


.91104 


39 


22 


.34803 


•93748 


.36434 


.93127 


.38053 


.92477 


.39661 


.91799 


.41257 


.91092 


38 


23 


.34830 


•93738 


.36461 


.93116 


.38080 


.92466 


.39688 


.91787 


.41284 


.91080 


37 


24 


.34857 


.93728 


.36488 


.93106 


.38107 


.92455 


.39715 


.91775 


.41310 


.91068 


36 


25 


.34884 


.93718 


.36515 


.93095 


.38134 


.92444 


•39741 


.91764 


.41337 


.91056 


35 


26 


.34912 


.93708 


.36542 


.93084 


.38161 


.92432 


.39768 


.91752 


.41363 


.91044 


34 


27 


.34939 


.93698 


.36569 


.93074 


.38188 


.92421 


.39795 


.91741 


.41390 


.91032 


33 


28 


.34966 


.93688 


.36596 


.93063 


.3821S 


.92410 


.39822 


.91729 


.41416 


.91020 


32 


29 


.34993 


.93677 


.36623 


.93052 


.38241 


.92399 


.39848 


.91718 


•41443 


.91008 


31 


30 


.35021 


.93667 


.36650 


.93042 


.38268 


.92388 


.39875 


.91706 


.41469 


.90996 


30 


31 


.35048 


.93657 


.36677 


.93031 


•38295 


•92377 


.39902 


.91694 


.41496 


.90984 


29 


32 


•35075 


.93647 


.36704 


.93020 


.38322 


.92366 


.39928 


.91683 


.41522 


.90972 


28 


33 


.35102 


.93637 


.36731 


.93010 


•38349 


.92355 


.39955 


.91671 


.41549 


.90960 


27 


34 


.35130 


.93626 


.36758 


.92999 


.38376 


.92343 


.39982 


.91660 


.41575 


.90948 • 


26 


35 


.35157 


.93616 


.36785 


.92988 


.38403 


.92332 


.40008 


.91648 


.41602 


.90936 


25 


36 


.35184 


.93606 


.36812 


.92978 


.38430 


.92321 


.40035 


.91636 


.41628 


.90924 


24 


37 


.35211 


.93596 


• 36839 


.92967 


.38456 


.92310 


.40062 


.91625 


.41655 


.90911 


23 


38 


.35239 


• 9358s 


• 36867 


.92956 


.38483 


.92299 


.40088 


.91613 


.41681 


.90899 


22 


39 


.35266 


•93575 


•36894 


•92945 


.38510 


.92287 


.40115 


.91601 


.41707 


.90887 


21 


40 


.35293 


•93565 


.36921 


.92935 


.38537 


.92276 


.40141 


.91590 


.41734 


.90875 


20 


41 


.35320 


•93555 


.36948 


.92924 


.38564 


.92265 


.40168 


.91578 


.41760 


.90863 


19 


42 


.35347 


•93544 


.36975 


.92913 


.38591 


.92254 


.40195 


.91566 


•41787 


.90851 


18 


43 


.35375 


•93534 


.37002 


.92902 


.38617 


.92243 


.40221 


.91555 


.41813 


.90839 


17 


44 


.35402 


•93524 


.37029 


.92892 


.38644 


.92231 


.40248 


•91543 


.41840 


.90826 


16 


45 


.35429 


.93514 


.37056 


.92881 


.38671 


.92220 


.40275 


•91531 


.41866 


.90814 


15 


46 


.35456 


.93503 


.37083 


.92870 


.3^698 


.92209 


.40301 


•91519 


.41892 


.90802 


14 


47 


.35484 


.93493 


.37110 


.92859 


.38725 


.92198 


.40328 


.91508 


.41919 


.90790 


13 


48 


.35511 


.93483 


.37137 


.92849 


.38752 


.92186 


.40355 


.91496 


.41945 


.90778 


12 


49 


.35538 


.93472 


.37164 


.92838 


.38778 


.92175 


.40381 


.91484 


.41972 


.90766 


11 


SO 


.35565 


•93462 


.37191 


.92827 


.38805 


.92164 


.40408 


.91472 


.41998 


.90753 


10 


51 


.35592 


.93452 


.37218 


.92816 


•38832 


.92152 


.40434 


.91461 


.42024 


.90741 


9 


52 


.35619 


•93441 


.37245 


.92805 


.38859 


.92141 


.40461 


.91449 


.42051 


.90729 


8 


S3 


.35647 


•93431 


.37272 


.92794 


.38886 


.92130 


.40488 


.91437 


.42077 


.90717 


7 


54 


.35674 


•93420 


•37299 


.92784 


.38912 


.92119 


.40514 


.91425 


.42104 


.90704 


6 


55 


.35701 


.93410 


.37326 


.92773 


•3S939 


.92107 


.40541 


.91414 


.42130 


.90692 


5 


56 


.35728 


.93400 


.37353 


.92762 


.38966 


.92096 


.40567 


.91402 


.42156 


.90680 


4 


^l 


.35755 


•93389 


.37380 


.92751 


.38993 


.92085 


.40594 


•91390 


.42183 


.90668 


3 


58 


.35782 


.93379 


.37407 


.92740 


.39020 


.92073 


.40621 


.91378 


.42209 


.90655 


2 


59 


.35810 


.93368 


.37434 


.92729 


.39046 


.92062 


.40647 


.91366 


.42235 


.90643 


I 


60 


.35837 


.93358 


.37461 


.92718 


.39073 


.92050 


.40674 


.91355 


.42262 


.90631 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


6( 


)° 


6^ 


B° 


6: 


7" 


6( 


3° 


6 


5° 



188 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



/ 


25 





26° 


27 





28 





29° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.42262 


.90631 


.43837 


.89879 


.45399 


.89101 


•46947 


.88295 


.48481 


.87462 


60 


I 


.42288 


.90618 


.43863 


.89867 


.45425 


.89087 


.46973 


.88281 


.48506 


.8:448 


59 


2 


.42315 


.90606 


.43889 


.89854 


.45451 


.89074 


.46999 


.88267 


.48532 


.87434 


S8 


3 


.42341 


.90594 


.43916 


.89841 


.45477 


.89061 


.47024 


.88254 


.48557 


.87420 


57 


4 


.42367 


.90582 


.43942 


.89828 


.45503 


.89048 


.47050 


.88240 


.48583 


.87406 


56 


S 


.42394 


.90569 


.43968 


.89816 


.45529 


■89035 


.47076 


.88226 


.48608 


.87391 


55 


6 


.42420 


.90557 


■ 43994 


.89803 


.45554 


.89021 


.47101 


.88213 


.48634 


.87377 


54 


7 


.42446 


.90545 


.44020 


.89790 


.45580 


.89008 


.47127 


.88199 


■48659 


.87363 


53 


8 


.42473 


.90532 


.44046 


.89777 


.45606 


.88995 


.47153 


.88185 


.48684 


.87349 


52 


9 


.42499 


.90520 


.44072 


.89764 


.45632 


.88981 


.47178 


.88172 


.48710 


•87335 


51 


10 


.42525 


.90507 


.44098 


.89752 


.45658 


.88968 


.47204 


.8815S 


•48735 


.87321 


50 


II 


.42552 


.90495 


.44124 


.89739 


.45684 


.88955 


•47229 


.88144 


.48761 


•87306 


49 


12 


.42578 


.90483 


.44151 


.89726 


■45710 


.88942 


•4725s 


.88130 


.48786 


.87292 


48 


13 


.42604 


.90470 


■44177 


.89713 


■45736 


.88928 


.47281 


.88117 


.48811 


.87278 


47 


14 


.42631 


.90458 


■44203 


.89700 


■45762 


.8891S 


.47306 


.88103 


.4S837 


.87264 


46 


15 


.42657 


.90446 


.44229 


.89687 


■45787 


.88902 


.47332 


.88089 


.48862 


.87250 


45 


i6 


.42683 


.90433 


■44255 


.89674 


.45813 


.88888 


.47358 


.88075 


.48888 


•8723s 


44 


ly 


.42709 


.90421 


.44281 


.89662 


.45839 


.88875 


.47383 


.88062 


.48913 


.87221 


43 


l8 


.42736 


.90408 


■44307 


.89649 


.45865 


.88862 


.47409 


.88048 


.48938 


.87207 


42 


19 


.42762 


.90396 


■44333 


.89636 


.45891 


.88848 


.47434 


.88034 


.48964 


•87193 


41 


20 


.42788 


.903S3 


.44359 


.89623 


.45917 


.88835 


.47460 


.88020 


.48989 


.87178 


40 


21 


.42815 


.90371 


•44385 


.89610 


■45942 


.88822 


.47486 


.88006 


.49014 


.87164 


39 


22 


.42841 


■90358 


.44411 


.89597 


■45968 


.88808 


.47511 


.87993 


.49040 


.87150 


38 


23 


.42867 


.90346 


.44437 


.89584 


•45994 


.88795 


.47537 


.87979 


.49065 


.87136 


37 


24 


.42894 


.90334 


.44464 


.89571 


.460-J 


.88782 


.47562 


.87965 


.49090 


.87121 


36 


25 


.42920 


■90321 


.44490 


.89558 


.46046 


.88768 


.47588 


.87951 


.49116 


.87107 


35 


26 


.42946 


.90309 


.44516 


.89545 


.46072 


.88755 


.47614 


.87937 


.49141 


.87093 


34 


27 


.42972 


.90296 


.44542 


.89532 


■46097 


.88741 


.47639 


.87923 


.49166 


.87079 


33 


28 


■42999 


.90284 


.44568 


.89519 


.46123 


.88728 


.47665 




.49192 


.87064 


32 


29 


.43025 


.90271 


.44594 


.89506 


.46149 


.88715 


.47690 


187896 


.49217 


.8701^0 


31 


30 


.43051 


.90259 


.44620 


.89493 


.46175 


.88701 


.47716 


.87882 


.49242 


.87036 


30 


31 


•43077 


.90246 


.44646 


.89480 


.46201 


.88688 


.47741 


.87868 


.49268 


.87021 


^9 


32 


.43104 


.90233 


.44672 


.89467 


.46226 


.88674 


.47767 


.87854 


•49293 


.87007 


28 


33 


.43130 


.90221 


.44698 


.89454 


.46252 


.88661 


.47793 


.87840 


•49318 


.86993 


27 


34 


.43156 


.90208 


.44724 


■89441 


.46278 


.88647 


.47818 


.87826 


•49344 


.86578 


26 


35 


.43182 


.90196 


.44750 


.89428 


■46304 


.88634 


.47844 


.87812 


•49369 


.86964 


25 


36 


.43209 


.90183 


.44776 


.89415 


■46330 


.88620 


.47869 


.87798 


.49394 


.86949 


24 


37 


.43235 


.90171 


.44802 


.89402 


■46355 


.88607 


.47895 


■87784 


.49419 


.86935 


23 


38 


.43261 


.90158 


.44828 


.89389 


■ 46381 


.88593 


.47920 


.87770 


.49445 


.86921 


22 


39 


.43287 


.90146 


.44854 


■89376 


■46407 


.88580 


.47946 


.87756 


.49470 


.86906 


21 


40 


•43313 


.90133 


.44880 


.89363 


.46433 


.88566 


.47971 


.87743 


.49495 


.86892 


20 


41 


.43340 


.90120 


.44906 


.89350 


.46458 


.88553 


.47997 


■87729 


.49521 


.86878 


19 


42 


.43366 


.90108 


■44932 


.89337 


.46484 


.88539 


.48022 


.87715 


•49546 


.86863 


18 


43 


.43392 


.90095 


■44958 


■89324 


.46510 


.88526 


.48048 


.87701 


.49571 


.86849 


17 


44 


.43418 


.90082 


■ 44984 


.89311 


.46536 


.88512 


.48073 


.87687 


.49596 


.86834 


16 


45 


.43445 


.90070 


.45010 


.89298 


.46561 


.88499 


.48099 


.87673 


.49622 


.86820 


15 


46 


.43471 


.90057 


.45036 


.89285 


.46587 


.88485 


.48124 


.87659 


.49647 


.86805 


14 


47 


•43497 


.90045 


.45062 


.89272 


.46613 


.88472 


.48150 


.87645 


.49672 


.86791 


13 


48 


.43523 


.90032 


■45088 


.89259 


.46639 


.88458 


.48175 


.87631 


•49697 


.86777 


12 


49 


•43549 


.90019 


■45114 


.89245 


.46664 


.88445 


.48201 


.87617 


.49723 


.86762 


II 


SO 


.43575 


.90007 


■45140 


.89232 


.46690 


.88431 


.48226 


.87603 


.40748 


.86748 


10 


51 


.43602 


.89994 


.45166 


.89219 


.46716 


.88417 


.48252 


.87589 


.49773 


.86733 


9 


52 


.43628 


.89981 


■45192 


.89206 


.46742 


.88404 


.48277 


.87575 


.49798 


.86719 


8 


53 


•43654 


.89968 


.45218 


.89193 


.46767 


.88390 


.48303 


.87561 


.49824 


.86704 


7 


54 


.43680 


.89956 


.45243 


.89180 


.46793 


.88377 


.48328 


.87546 


.49849 


.86690 


6 


55 


.43706 


.8y943 


.45269 


.89167 


.46819 


.88363 


■48354 


.87532 


.49874 


.86675 


s 


S6 


•43733 


.89930 


.45295 


.89153 


.46844 


.88349 


■48379 


.87518 


.49899 


.86661 


4 


57 


•43759 


.89918 


.45321 


.89140 


.46870 


.88336 


.48405 


.87504 


.49924 


.86646 


3 


58 


.43785 


.89905 


.45347 


.89127 


.46896 


.88322 


.48430 


.87490 


.49950 


.86632 


2 


59 


.43811 


.89892 


.45373 


.89114 


.46921 


.88308 


.48456 


.87476 


.49975 


.86617 


I 


6o 


.43837 


.89879 


.45399 


.89101 


.46947 


.88295 


.48481 


.87462 


.50000 


.86603 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


6. 


i° 


6 


3° 


62 





61 





6 


0° 



189 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



/ 


30° 


3 


1° 


3- 


2° 


33° 


34° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.50000 


.86603 


.51504 


.85717 


.52992 


.84805 


.54464 


.83867 


.55919 


.82904 


60 


I 


.50025 


.86588 


.51529 


.85702 


.53017 


.84789 


.54488 


.83851 


.55943 


.82887 




2 


.50050 


.86573 


.51554 


.85687 


.53041 


.84774 


.54513 


.83835 


.55968 


.82871 


58 


3 


.50076 


.86559 


.51579 


.85672 


.53066 


.84759 


.54537 


.83819 


•55992 


.82855 


57 


4 


.50101 


.86544 


.51604 


.85657 


.53091 


.84743 


.54561 


.83804 


.56016 


.82839 


56 


s 


.50126 


.86530 


.51628 


.85642 


.53115 


.84728 


.54586 


.83788 


.56040 


.82822 


55 


6 


.50151 


.86515 


.51653 


.85627 


.53140 


.84712 


.54610 


.83772 


.56064 


.82806 


54 


7 


.50176 


.86501 


.S1678 


.856x2 


.53164 


.84697 


.54635 


.83756 


.56088 


.82790 


53 


8 


.50201 


.86486 


.51703 


.85597 


.53189 


.84681 


.54659 


.83740 


.56112 


•^2773 


52 


9 


.50227 


.86471 


.51728 


.85582 


.53214 


.84666 


.54683 


.83724 


.56136 


.82757 


51 


10 


.50252 


.86457 


.51753 


.85567 


.53238 


.84650 


.54708 


.83708 


.56160 


.82741 


SO 


II 


.50277 


.86442 


.51778 


.85551 


.53263 


.84635 


.54732 


.83692 


.56184 


.82724 


49 


12 


.50302 


.86427 


.51803 


.85536 


.53288 


.84619 


.54756 


.83676 


.56208 


.82708 


48 


13 


.50327 


.86413 


.51828 


.85521 


.53312 


.84604 


.54781 


.83660 


.56232 


.82692 


47 


14 


.50352 


.86398 


.51852 


.85506 


.53337 


.84588 


.54805 


.83645 


.56256 


.82675 


46 


IS 


.50377 


.86384 


.51877 


.85491 


.53361 


.84573 


.54829 


.83629 


.56280 


.82659 


45 


i6 


.50403 


.86369 


.51902 


.85476 


.53386 


.84557 


.54854 


.83613 


.56305 


.82643 


44 


17 


.50428 


.86354 


.51927 


.85461 


.53411 


.84542 


.54878 


.83597 


.56329 


.82626 


43 


i8 


.50453 


.86340 


.51952 


.85446 


.53435 


.84526 


.54902 


.83581 


.56353 


.82610 


42 


19 


.50478 


.86325 


.51977 


.85431 


.53460 


.84511 


.54927 


.83565 


.56377 


.82593 


41 


20 


.50503 


.86310 


.52002 


.85416 


.53484 


.84495 


.54951 


.83549 


.56401 


.82577 


40 


21 


.50528 


.86295 


.52026 


.85401 


.53509 


.84480 


.5497s 


.83533 


.56425 


.82561 


39 


22 


.50553 


.86281 


.52051 


.85385 


.53534 


.84464 


.54999 


.83517 


.56449 


•82544 


38 


23 


.50578 


.86266 


.52076 


.85370 


.53558 


.84448 


.55024 


.83501 


.56473 


.82528 


37 


24 


.50603 


.86251 


.52101 


.85355 


.53583 


.84433 


.55048 


.83485 


.56497 


.82511 


36 


25 


.50628 


.86237 


.52126 


.85340 


.53607 


.84417 


.55072 


.83469 


.56521 


.82495 


35 


26 


.50654 


.86222 


.52151 


.85325 


.53632 


.84402 


.55097 


.83453 


.56545 


.82478 


34 


27 


.50679 


.86207 


.52175 


.85310 


.53656 


.84386 


.55121 


.83437 


.56569 


.82462 


33 


28 


.50704 


.86192 


.52200 


.85294 


.53681 


.84370 


.55145 


.83421 


.56593 


.82446 


32 


29 


.50729 


.86178 


.52225 


.85279 


.53705 


.84355 


.55169 


.83405 


•56617 


.82429 


31 


30 


.50754 


.86163 


.52250 


.85264 


.53730 


.84339 


.55194 


.83389 


.56641 


•82413 


30 


31 


.50779 


.86148 


.52275 


.85249 


.53754 


.84324 


.55218 


.83373 


.56665 


.82396 


29 


32 


.50804 


.86133 


.52299 


.85234 


.53779 


.84308 


•55242 


.83356 


.56689 


.82380 


28 


33 


.50829 


.86119 


.52324 


.85218 


.53804 


.84292 


.55266 


.83340 


.56713 


.82363 


27 


34 


.50854 


.86104 


.52349 


.85203 


.53828 


.84277 


.55291 


.83324 


.56736 


.82347 ■ 


26 


35 


.50879 


.86089 


.52374 


.85188 


.53853 


.84261 


.55315 


.83308 


.56760 


.82330 


25 


36 


.50904 


.86074 


.52399 


.85173 


.53877 


.84245 


.55339 


.83292 


.56784 


.82314 


24 


37 


.50929 


.86059 


.52423 


.85157 


.53902 


.84230 


.55363 


.83276 


.56808 


.82297 


23 


38 


.50954 


.86045 


.52448 


.85142 


.53926 


.84214 


.55388 


.83260 


.56832 


.82281 


22 


39 


.50979 


.86030 


.52473 


.85127 


.53951 


.84198 


.55412 


.83244 


.56856 


.82264 


21 


40 


.51004 


.86015 


.52498 


.85112 


.53975 


.84182 


.55436 


.83228 


-56880 


.82248 


20 


41 


.51029 


.86000 


.52522 


.85096 


.54000 


.84167 


.55460 


.83212 


.56904 


.82231 


19 


42 


.51054 


.85985 


.52547 


.85081 


.54024 


.84151 


.55484 


.83195 


.56928 


.82214 


18 


43 


.51079 


.85970 


.52572 


.85066 


.54049 


.84135 


.55509 


.83179 


.56952 


.82198 


17 


44 


.51104 


.85956 


.52597 


.85051 


.54073 


.84120 


.55533 


.83163 


.56976 


.82181 


16 


45 


.51129 


.85941 


.52621 


.85035 


.54097 


.84104 


.55557 


.83147 


.57000 


.82165 


IS 


46 


.51154 


.85926 


.52646 


.85020 


.54122 


.84088 


.55581 


.83131 


.57024 


.82148 


14 


47 


.51179 


.85911 


.52671 


.85005 


.54146 


.84072 


.55605 


.83115 


.57047 


.82132 


13 


48 


.51204 


.85896 


.52696 


.84989 


.54171 


.84057 


.55630 


.83098 


.57071 


.82115 


12 


49 


.51229 


.85881 


.52720 


.84974 


.54195 


.84041 


.55654 


.83082 


.57095 


.82098 


II 


SO 


.51254 


.85866 


.52745 


.84959 


.54220 


.84025 


.55678 


.83066 


.57119 


.82082 


10 


51 


.51279 


.85851 


.52770 


.84943 


.54244 


.84009 


.55702 


.83050 


.57143 


.82065 


9 


52 


.51304 


.85836 


.52794 


.84928 


.54269 


.83994 


.55726 


.83034 


.57167 


.82048 


8 


S3 


.51329 


.85821 


.52819 


.84913 


.54293 


.83978 


.55750 


.83017 


.57191 


.82032 


7 


54 


.51354 


.85806 


.52844 


.84897 


.54317 


.83962 


.55775 


.83001 


.57215 


.82015 


6 


SS 


.51379 


.85792 


.52869 


.84882 


.54342 


.83946 


.55799 


.82985 


.57238 


.81999 


S 


S6 


.51404 


.85777 


.52893 


.84866 


.54366 


.83930 


.55823 


.82969 


.57262 


.81982 


4 


57 


.51429 


.85762 


.52918 


.84851 


.54391 


.83915 


.55847 


.82953 


.57286 


.81965 


3 


58 


.51454 


.85747 


.52943 


.84836 


.54415 


.83899 


.55871 


.82936 


.57310 


.81949 


2 


59 


.51479 


.85732 


.52967 


.84820 


.54440 


.83883 


.55895 


.82920 


.57334 


.81932 


I 


60 


.51504 


.85717 


.52992 


.84805 


.54464 


.83867 


.55919 


.82904 


.57358 


.81915 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


59 


)° 


5^ 


5° 


5/ 


7O 


si 


i° 


5f 


.0 

5 



190 



BROWN & SHARPE MFG. CO. 

NATURAL SINES AND COSINES 



f 


35° 


36 





-2*7° 

37 


38° 


39° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


.57358 


.81915 


.58779 


.80902 


.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


60 


1 


.57381 


.81899 


.58802 


.80885 


.60205 


• 79846 


.61589 


.78783 


.62955 


.77696 


59 


2 


.57405 


.81882 


.58826 


.80867 


.60228 


.79829 


.61612 


.78765 


.62977 


.77678 


58 


3 


• 57429 


.81865 


•58849 


.80850 


.60251 


•79811 


.61635 


.78747 


.63000 


.77660 


57 


4 


.57453 


.81848 


.58873 


.80833 


.60274 


•79793 


.61658 


.78729 


.63022 


.77641 


56 


S 


.57477 


.81832 


.58896 


.80816 


.60298 


.79776 


.61681 


.78711 


•6304s 


.77623 


55 


6 


.57501 


.8181S 


.58920 


•80799 


.60321 


•79758 


.61704 


.78694 


.63068 


.77605 


54 


7 


.57524 


.81708 


.58943 


.80782 


.60344 


.79741 


.61726 


.78676 


.63090 


.77586 


53 


8 


.57548 


.81782 


.58967 


.80765 


.60367 


.79723 


.61749 


.78658 


.63113 


.77568 


52 


9 


.57572 


.81765 


.58990 


.80748 


.60390 


.79706 


.61772 


.78640 


.63135 


.77550 


51 


10 


.57596 


.81748 


.59014 


.80730 


.60414 


.79688 


•6179s 


.78622 


.63158 


.77531 


50 


II 


.57619 


.81731 


.59037 


.80713 


•60437 


.79671 


.61818 


.78604 


.63180 


.77513 


49 


12 


.57643 


.81714 


.59061 


.80696 


.60460 


.79653 


.61841 


.78586 


.63203 


.77494 


48 


13 


.57667 


.81698 


.59084 


.80679 


.60483 


.79635 


.61864 


.78568 


.63225 


.77476 


47 


14 


.57691 


.81681 


•59108 


.80662 


.60506 


.79618 


.61887 


.78550 


.63248 


.77458 


46 


15 


.57715 


.81664 


•59131 


.80644 


.60529 


.79600 


.61909 


.78532 


.63271 


• 77439 


45 


i6 


.57738 


.81647 


•59154 


.80627 


•60553 


.79583 


.61932 


.78514 


.63293 


•77421 


44 


17 


.57762 


.81631 


•59178 


.80610 


•60576 


.79565 


.61955 


.78496 


.63316 


• 77402 


43 


i8 


.57786 


.81614 


.59201 


•80593 


.60599 


.79547 


.61978 


.78478 


.633.^8 


.77384 


42 


19 


.57810 


.81597 


.59225 


•80576 


.60622 


.79530 


.62001 


.78460 


.63361 


.77366 


41 


20 


.57833 


.81580 


.59248 


.80558 


.60645 


.79512 


.62024 


.78442 


.63383 


.77347 


40 


21 


.57857 


.81563 


.59272 


•80541 


.60668 


.79494 


.62046 


.78424 


.63406 


.77329 


39 


22 


.57881 


.81546 


.59295 


.80524 


.60691 


.79477 


.62069 


.78405 


.63428 


•77310 


38 


23 


.57904 


.81530 


.59318 


.80507 


.60714 


.79459 


.62092 


.78387 


.63451 


•77292 


37 


24 


.57928 


.81513 


.59342 


.80489 


.60738 


.79441 


.62115 


.78369 


.63473 


.77273 


36 


25 


.57952 


.81496 


.59365 


.80472 


.60761 


.79424 


.62138 


.78351 


.63496 


•77255 


35 


26 


.57976 


.81479 


• 59389 


•80455 


.60784 


.79406 


.62160 


■ 7^323 


•63518 


• 77236 


34 


27 


.57999 


.81462 


•59412 


.80438 


.60807 


.79388 


.62183 


.78315 


•63540 


.77218 


33 


28 


.58023 


.81445 


•59436 


.80420 


.60830 


.79371 


.62206 


.78297 


•63563 


.77199 


32 


29 


.58047 


.81428 


•59459 


.80403 


.60853 


.79353 


.62229 


.78279 


•63585 


•77181 


31 


30 


.58070 


.81412 


.59482 


.80386 


.60876 


-.79335 


.62251 


.78261 


.63608 


.77162 


30 


31 


.58094 


.81395 


•59506 


.80368 


.60899 


.79318 


.62274 


.78243 


.63630 


• 77144 


29 


32 


.58118 


.81378 


•59529 


.80351 


.60922 


.79300 


.62297 


.78225 


•63653 


.77125 


28 


33 


.58141 


.81361 


• 595*52 


•80334 


.60945 


.79282 


.62320 


.78206 


•63675 


.77107 


27 


34 


.58165 


.81344 


• 59576 


.80316 


.60968 


.79264 


.62342 


.78188 


.63698 


.77088 


26 


35 


.58189 


.81327 


•59599 


.80299 


.60991 


.79247 


.62365 


.78170 


.63720 


.77070 


25 


36 


.58212 


.81310 


.59622 


,80282 


.61015 


.79229 


.62388 


.78152 


.63742 


.77051 


24 


37 


.58236 


.81293 


.59646 


.80264 


.61038 


.79211 


.62411 


.78134 


.63765 


.77033 


23 


38 


.58260 


.81276 


•59669 


.80247 


.61061 


.79193 


.62433 


.78116 


.63787 


.77014 


22 


39 


.58283 


.81259 


• 59693 


.80230 


.61084 


,79176 


.62456 


.78098 


.63810 


.76996 


21 


40 


.58307 


.81242 


.59716 


.80212 


.61107 


.79158 


.62479 


.78079 


•63832 


.76977 


20 


41 


.58330 


.81225 


•59739 


.80195 


.61130 


.79140 


.62502 


.78061 


•63854 


• 76959 


19 


42 


.58354 


.81208 


• 59763 


.80178 


.61153 


.79122 


.62524 


.78043 


•63877 


•76940 


18 


43 


.58378 


.81191 


• 59786 


.80160 


.61176 


.79105 


.62547 


.78025 


.63899 


.76921 


17 


44 


.58401 


.81174 


.59809 


•80143 


.61199 


.79087 


.62570 


.78007 


.63922 


.76903 


16 


45 


.58425 


.81157 


.59832 


.80125 


.61222 


.79069 


.62592 


.77988 


.63944 


.76884 


15 


46 


.58449 


.81140 


.59856 


.80108 


.61245 


.79051 


.62615 


.77970 


.63966 


.76866 


14 


47 


.58472 


.81123 


•59879 


.80091 


.61268 


.79033 


.62638 


.77952 


.63989 


• 76847 


13 


48 


.58496 


.81106 


.59902 


.80073 


.61291 


.79016 


.62660 


.77934 


.64011 


.76828 


12 


49 


.58519 


.81089 


.59926 


.80056 


.61314 


.78998 


.62683 


.77916 


.64033 


.76810 


11 


50 


.58543 


.81072 


•59949 


.80038 


.61337 


.78980 


.62706 


.77897 


.64056 


.76791 


10 


51 


.58567 


.810SS 


• 59972 


.80021 


.61360 


.78962 


.62728 


.77879 


.64078 


.76772 


9 


52 


.58590 


.81038 


•59995 


.80003 


.61383 


.78944 


.62751 


.77861 


.64100 


.76754 


8 


53 


.58614 


.81021 


.60019 


.79986 


.61406 


.78926 


.62774 


.77843 


.64123 


.76735 


7 


54 


•S^^F 


.81004 


.60042 


.79968 


.61429 


.78908 


.62796 


.77824 


.64145 


.76717 


6 


55 


.58661 


.80987 


.60065 


•79951 


.61451 


.78891 


.62819 


.77806 


.64167 


.76698 


5 


S6 


.58684 


.80970 


.60089 


• 79934 


•61474 


.78873 


.62842 


.77788 


.64190 


.76679 


4 


^l 


.58708 


•80953 


.60112 


.79916 


•61497 


.78855 


.62864 


.77769 


.64212 


.76661 


3 


S8 


.58731 


.80936 


.60135 


.79899 


.61520 


.78837 


.62887 


.77751 


.64234 


.76642 


2 


59 


.58755 


.80919 


.60158 


.79881 


•61543 


.78819. 


.62909 


.77733 


.64256 


.76623 


I 


60 


.58779 


.80902 


.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


.64279 


.76604 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


5^ 


t° . 


53 





5i 


>° 


51 


° 


5< 


3° 



191 



BROWN & SHARPE MFG. CO. 
NATURAL SINES AND COSINES 



f 


40 


° 


41 





42° 


43° 


44° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.64279 


.76604 


.65606 


.75471 


.66^13 


.74314 


.68200 


.73135 


.69466 


.71934 


60 


I 


.64301 


.76586 


.65628 


.75452 


.66935 


.74295 


.68221 


.73116 


.69487 


.71914 


59 


2 


.64323 


.76567 


.65650 


.75433 


.66956 


.74276 


.68242 


.73096 


.69508 


.71894 


58 


3 


.64346 


.76548 


.65672 


.75414 


.66978 


.74256 


.68264 


.73076 


.69529 


.71873 


57 


4 


.64368 


.76530 


.65694 


.75395 


.66999 


.7^23,7 


.68285 


.73056 


.69549 


.71853 


56 


S 


.64390 


.76511 


.65716 


.75375 


.67021 


.74217 


.68306 


.73036 


•69570 


.71833 


55 


6 


.64412 


.76492 


.65738 


.75356 


.67043 


.74198 


•68327 


.73016 


•69591 


.71813 


54 


7 


.64435 


.76473 


.65759 


.■7S337 


.67064 


.74178 


.68349 


.72996 


.69612 


.71792 


S3 


8 


.64457 


•76455 


.65781 


.75318 


.67086 


.74159 


.68370 


.72976 


.69633 


.71772 


52 


9 


.64479 


•76436 


.65803 


.75299 


.67107 


.74139 


.68391 


.72957 


•69654 


.71752 


51 


10 


.64501 


.76417 


.65825 


.75280 


.67129 


.74120 


,68412 


■729Z7 


•6967s 


.71732 


50 


II 


.64524 


.76398 


.65847 


.75261 


.67151 


.74100 


.68434 


.72917 


.69696 


.71711 


49 


12 


.64546 


.76380 


.65869 


.75241 


.67172 


.74080 


.68455 


.72897 


.69717 


.71691 


48 


13 


.64568 


.76361 


.65891 


.75222 


.67194 


.74061 


.68476 


.72877 


•69737 


.71671 


47 


14 


.64590 


.76342 


.65913 


.75203 


.67215 


.74041 


.68497 


.72857 


.69758 


.71650 


46 


IS 


.64612 


.76323 


.65935 


.75184 


.67237 


.74022 


.68518 


.72837 


.69779 


.71630 


45 


i6 


.64635 


.76304 


.65956 


.75165 


.67258 


.74002 


.68539 


.72817 


.69800 


.71610 


44 


17 


.64657 


.76286 


.65978 


.75146 


.67280 


.73983 


.68561 


.72797 


.69821 


.71590 


43 


i8 


.64679 


.76267 


.66000 


.75126 


.67301 


.73963 


.68582 


.72777 


.6 842 


.71569 


42 


19 


.64701 


.76248 


.66022 


.75107 


.67323 


.73944 


.68603 


.72757 


.69:62 


.71549 


41 


20 


.64723 


.76229 


.66044 


.75088 


.67344 


.73924 


.68624 


.72737 


.69883 


.71529 


40 


21 


.64746 


.76210 


.66066 


.75069 


.67366 


.73904 


.68645 


.72717 


.69904 


.71508 


39 


22 


.64768 


.76192 


.66088 


•75050 


.67387 


.73885 


.68666 


.72697 


.69925 


.71488 


38 


23 


.64790 


.76173 


.66109 


.75030 


.67409 


.7386s 


.68688 


.72677 


.69946 


.71468 


37 


24 


.64812 


.76154 


.66131 


.75011 


.67430 


.73846 


.68709 


.72657 


.69966 


.71447 


36 


25 


.64834 


.76135 


.66153 


.74992 


.67452 


.73826 


.68730 


.72637 


.69987 


.71427 


35 


26 


.64856 


.76116 


.66175 


.74973 


.67473 


.73806 


.6C751 


.72617 


.70008 


.71407 


34 


27 


.64878 


.76097 


.66197 


.74953 


.67495 


•73787 


.6Gr72 


.72597 


.70029 


.71386 


33 


28 


.64901 


.76078 


.66218 


.74934 


.67516 


•73767 


.68793 


.72577 


.70049 


.71366 


32 


29 


.64923 


.76059 


.66240 


.74915 


.67538 


.73747 


.68C14 


.72557 


.70070 


.71345 


31 


30 


.64945 


.76041 


.66262 


.74896 


.67559 


.73728 


.C8C3S 


.72537 


.70091 


.71325 


30 


31 


.64967 


.76022 


.66284 


.74876 


.67580 


.73708 


.68857 


.72517 


.70112 


.71305 


29 


32 


.64989 


.76003 


.66306 


.74857 


.67602 


.73688 


.68878 


.72497 


.70132 


.71284 


28 


33 


.65011 


.75984 


.66327 


.74838 


.67623 


.73669 


.68899 


.72477 


.70153 


.71264 • 


27 


34 


.65033 


.75965 


.66349 


.74818 


.67645 


.73649 


.68920 


.72457 


.70174 


.71243 


26 


35 


.65055 


.75946 


.66371 


.74799 


.67666 


.73629 


.68941 


.72437 


.70195 


.71223 


25 


36 


.65077 


.75927 


.66393 


.74780 


.67688 


.73610 


.68962 


.72417 


.70215 


.71203 


24 


37 


.65100 




.66414 


.74760 


.67709 


.73590 


.68983 


.72397 


.70236 


.71182 


23 


38 


.65122 


.75889 


.66436 


•74741 


.67730 


.73570 


.69004 


.72377 


.70257 


.71162 


22 


39 


.65144 


.75870 


.66458 


.74722 


.67752 


.73551 


.69025 


.72357 


.70277 


.71141 


21 


40 


.65166 


.75851 


.66480 


■74703 


.67773 


.73531 


.69046 


.72337 


.70298, 


.71121 


20 


41 


.65188 


.75832 


.66501 


.74683 


.67795 


.73511 


.69067 


.72317 


.70319 


.71100 


19 


42 


.65210 


.75813 


.66523 


.74664 


.67816 


.73491 


.69088 


.72297 


.70339 


.71080 


18 


43 


.65232 


.75794 


•66545 


.74644 


.67837 


.73472 


.69109 


.72277 


.70360 


.71059 


17 


44 


.65254 


.75775 


.66566 


.74625 


.67859 


.73452 


.69x30 


.72257 


.70381 


.71039 


16 


45 


.65276 


.75756 


.66588 


.74606 


.67880 


.73432 


.69151 


.72236 


.70401 


.71019 


IS 


46 


.65298 


.75738 


66610 


.74586 


.67901 


.73413 


.69172 


.72216 


.70422 


.70998 


14 


47 


.65320 


.75719 


.66632 


.74567 


.67923 


.73393 


.69193 


.72196 


.70443 


.70978 


13 


48 


.65342 


.75700 


.66653 


.74548 


.67944 


.73373 


.69214 


.72176 


.70463 


.70957 


12 


49 


.65364 


.75680 


.66675 


.74528 


.67965 


.73353 


.69235 


.72156 


.70484 


.70937 


11 


SO 


.65386 


.75661 


.66697 


.74509 


.67987 


.73333 


.69256 


.72136 


.70505 


.70916 


10 


51 


.65408 


.75642 


.66718 


.74489 


.68008 


.73314 


.69277 


.72116 


.70525 


.70896 


9 


52 


.65430 


.75623 


.66740 


.74470 


.68029 


.73294 


.69298 


.72095 


.70546 


.70875 


8 


53 


.65452 


.75604 


.66762 


.74451 


.68051 


.73274 


•69319 


.72075 


.70567 


.70855 


7 


54 


.65474 


.75585 


.66783 


.74431 


.68072 


.73254 


.69340 


.72055 


.70587 


.70834 


6 


55 


.65496 


.75566 


.66805 


.74412 


.68093 


.73234 


.69361 


.72035 


.70608 


.70813 


s 


56 


.65518 


.75547 


.66827 


.74392 


.68115 


.73215 


.69382 


.72015 


.70628 


.70793 


4 


57 


.65540 


.75528 


.66848 


.74373 


.68136 


.73195 


.69403 


.71995 


.70649 


.70772 


3 


58 


.65562 


.75509 


.66870 


.74353 


.68157 


.73175 


•69424 


.71974 


.70670 


.70752 


2 


59 


.65584 


.75490 


.66891 


.74334 


.68179 


.73155 


.69445 


.71954 


.70690 


.70731 


I 


60 


.65606 


•7S47I 


.66913 


.74314 


.68200 


.73135 


.69466 


.71934 


.70711 


,70711 





f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


4< 


f 


4^ 


?° 


4' 


7° 


4< 


5° 


4 


5° 



192 



Natural Tangents and Cotangents 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTS AND COTANGENTS 



/ 


0° 


1° 


2 





3 





4° 


/ 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


Cotang: 





.00000 


Infinite 


.01746 


57.2900 


.03492 


28.6363 


.05241 


19.0811 


.06993 


14.3007 


60 


I 


.00029 


3437.75 


.01775 


56.3506 


.03521 


28.3994 


.05270 


18.9755 


.07022 


14.2411 


S9 


2 


.00058 


1718.87 


.01804 


55.4415 


.03550 


28.1664 


.05299 


18.8711 


.07051 


14.1821 


S8 


3 


.00087 


1145.92 


.01833 


54.5613 


-03579 


27.9372 


.05328 


18.7678 


.07080 


14.123s 


57 


4 


.00116 


859.436 


.01862 


5'3.7o86 


.03609 


27.7117 


.05357 


18.6656 


.07110 


14.0655 


S6 


5 


.00145 


687.549 


.01891 


52.8821 


.03638 


27.4899 


.05387 


18.5645 


.07139 


14.0079 


55 


6 


.00175 


572.957 


.01920 


52.0807 


.03667 


■27.271s 


.05416 


18.4645 


.07168 


13.9507 


54 


7 


.00204 


491.106 


.01949 


S1.3032 


.03696 


27.0566 


.05445 


18.3655 


.07197 


13.8940 


53 


8 


.00233 


429.718 


.01978 


50.5485 


.03725 


26.8450 


.05474 


18.2677 


.07227 


13.8378 


52 


9 


.00262 


381.971 


.02007 


49.8157 


.03754 


26.6367 


.05503 


18.1708 


.07256 


13.7821 


51 


10 


.00291 


343.774 


.02036 


49.1039 


.03783 


26.4316 


.05533 


18.0750 


.07285 


13.7267 


so 


II 


.00320 


312.521 


.02066 


48.4121 


.03812 


26.2296 


.05562 


17.9802 


.07314 


13.6719 


49 


12 


.00349 


286.478 


.02095 


47.7395 


.03842 


26.0307 


.05591 


17.8863 


.07344 


13.6174 


48 


13 


.00378 


264.441 


.02124 


47.0853 


.03871 


25.8348 


.05620 


17.7934 


■OIZIZ 


13.5634 


47 


14 


-00407 


245.552 


.02IS3 


46.4489 


.03900 


25.6418 


.05649 


17.7015 


.07402 


13.5098 


46 


IS 


.00436 


229.182 


.02182 


45.8294 


.03929 


25-4517 


.05678 


17.6106 


.07431 


13.4566 


45 


i6 


.00465 


214.858 


.02211 


45.2261 


.03958 


25-2644 


.05708 


17.520S 


.07461 


13.4039 


44 


17 


.00495 


202.219 


.02240 


44.6386 


.03987 


25-0798 


.05737 


17.4314 


.07490 


13.3515 


43 


i8 


.00524 


190.984 


.02269 


44.0661 


.04016 


24-8978 


.05766 


17.3432 


.07519 


13.2996 


42 


19 


.00553 


180.932 


.02298 


43.5081 


.04046 


24-7185 


.05795 


17.2558 


.07548 


13.2480 


41 


20 


.00582 


171.88s 


.02328 


42.9641 


.04075 


24-5418 


.05824 


17.1693 


.07578 


13.1969 


40 


21 


.00611 


163.700 


.02357 


42.4335 


.04104 


24.3675 


.05854 


17.0837 


.07607 


13.1461 


39 


22 


.00640 


156.259 


.02386 


41.9158 


.04133 


24.1957 


.05883 


16.9990 


.07636 


13.0958 


38 


23 


.00669 


149-465 


.0241S 


41.4106 


,04162 


24.0263 


.05912 


16.9150 


.07665 


13.0458 


37 


24 


.00698 


143.237 


.02444 


40.9174 


.04191 


23.8593 


.05941 


16.8319 


.07695 


12.9962 


36 


25 


.00727 


137.507 


.02473 


40.4358 


.04220 


23.6945 


.05970 


16.7496 


.07724 


12.9469 


35 


26 


.00756 


132.219 


.02502 


39.9655 


.04250 


23-5321 


.05999 


16.6681 


.07753 


12.8981 


34 


21 


.00785 


127.321 


.02531 


39.5059 


.04279 


23-3718 


.06029 


16.5874 


.07782 


12.8496 


33 


28 


.008 IS 


122.774 


.02560 


39.0568 


.04308 


23-2137 


.06058 


16.5075 


.07812 


12.8014 


32 


29 


.00844 


118.540 


.02589 


38.6177 


.04337 


23-0577 


.06087 


16.4283 


.07841 


12.7536 


31 


30 


.00873 


114.589 


,02^19 


38.188s 


.04366 


22.9038 


.06116 


16.3499 


.07870 


12.7062 


30 


31 


.00902 


110.892 


.02648 


37.7686 


.04395 


22.7519 


.06145 


16.2722 


-07899 


12.6591 


29 


32 


.00931 


107.426 


.02677 


37-3579 


.04424 


22.6020 


.06175 


16.1952 


.07929 


12.6124 


28 


33 


.00960 


104.171 


.02706 


36-9560 


.04454 


22.4541 


.06204 


16.1190 


.07958 


12.5660 


. 27 


34 


.00989 


101.107 


.02735 


36-5627 


.04483 


22,3081 


.06233 


16.0435 


.07987 


12.5199 


26 


35 


.01018 


98.2179 


.02764 


36-1776 


.04512 


22.1640 


,06262 


15.9687 


.08017 


12.4742 


25 


36 


.01047 


95.4895 


.02793 


35.8006 


.04541 


22.0217 


,06291 


15.8945 


.08046 


12.4288 


24 


37 


.01076 


92.908s 


.02822 


35.4313 


.04570 


21.8813 


.06321 


15.8211 


.08075 


12.3838 


23 


38 


.Olios 


90.4633 


.02851 


35-0695 


.04S99 


21,7426 


.06350 


15.7483 


.08104 


12.3390 


22 


39 


.01135 


88.1436 


.02881 


34-7151 


.04628 


21.6056 


.06379 


15.6762 


.08134 


12.2946 


21 


40 


.01164 


85.9398 


.02910 


34.3678 


.04658 


21.4704 


.06408 


15.6048 


.08163 


12.250s 


20 


41 


.01193 


83.843s 


.02939 


34.0273 


.04687 


21.3369 


.06437 


15.5340 


.08192 


12.2067 


19 


42 


.01222 


81.8470 


.02968 


33.6935 


.04716 


21.2049 


.06467 


1S.4638 


.08221 


12.1632 


18 


43 


.01251 


79.9434 


.02997 


33.3662 


.04745 


21.0747 


.06496 


15.3943 


.08251 


12.1201 


17 


44 


.01280 


78.1263 


.03026 


33.0452 


.04774 


20.9460 


.06525 


15.3254 


.08280 


12.0772 


16 


45 


.01309 


76.3900 


.03055 


32.7303 


.04803 


20.8188 


.06554 


15.2571 


.08309 


12.0346 


IS 


46 


.01338 


74.7292 


.03084 


32.4213 


.04833 


20.6932 


.06584 


1S.1893 


.08339 


11,9923 


14 


47 


.01367 


73.1390 


.03114 


32.1181 


.04862 


20.5691 


.06613 


15.1222 


.08368 


IX. 9504 


13 


48 


.01396 


71.6151 


.03143 


31.8205 


.04891 


20.4465 


,06642 


15.0557 


.08397 


11.9087 


12 


49 


.01425 


70.1533 


.03172 


31.5284 


.04920 


20.3253 


.06671 


14.9898 


.08427 


11.8673 


II 


SO 


.OI4SS 


68.7501 


.03201 


31.2416 


.04949 


20.2056 


.06700 


14.9244 


,08456 


11.826-2 


10 


SI 


.01484 


67.4019 


.03230 


30.9599 


.04978 


20.0872 


.06730 


14.8596 


,08485 


11.7853 


9 


52 


.01513 


66. loss 


.03259 


30.6833 


.05007 


19.9702 


.06759 


14-7954 


.08514 


11.7448 


8 


S3 


• .01542 


64.8580 


.03288 


30.4116 


.05037 


19.8546 


.06788 


14.7317 


.08544 


11.7045 


7 


54 


.01571 


63.6567 


•03317 


30.1446 


,05066 


19.7403 


.06817 


14.668s 


.08573 


11.6645 


6 


SS 


.01600 


62.4992 


.03346 


29.8823 


.05095 


19.6273 


.06847 


14.6059 


.08602 


11.6248 


s 


S6 


.01629 


61.3829 


.03376 


29.6245 


.05124 


19.5156 


.06876 


14.5438 


.08632 


11,5853 


4 


S7 


.01658 


60.3058 


.03405 


29.3711 


.05153 


19.4051 


.0690s 


14.4823 


,08661 


11,5461 


3 


58 


.01687 


59.2659 


.03434 


29.1220 


.05182 


19.2959 


.06934 


14.4212 


.08690 


11.5072 


2 


59 


.01716 


58.2612 


.03463 


28.8771 


.05212 


19.1879 


.06963 


14.3607 


.08720 


11.468s 


I 


6o 


,01746 


57.2900 


.03492 


28.6363 


.05241 


19.0811 


.06993 


14.3007 


.08749 


11.4301 





/ 


Cotang 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


' 


89° 


88° 


8: 


7° 


8( 


)° 


8 


-0 
3 



194 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTS AND COTANGENTS 



1 


5° 


6 


° 


7° 


8 





9° 


/ 


Tang: 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.08749 


11.4301 


.10510 


9.51436 


.12278 


8.14435 


.14054 


7.11537 


.15838 


6-31375 


60 


I 


.08778 


11-3919 


.10540 


9.48781 


.12308 


8.12481 


.14084 


7.10038 


.15868 


6.30189 


59 


2 


.08807 


11.3540 


.10569 


9.46141 


.12338 


8.10536 


.14113 


7.08546 


.15898 


6.29007 


58 


3 


.08837 


II. 3163 


.10599 


9.43515 


.12367 


8.08600 


• 14143 


7^07059 


.15928 


6.27829 


57 


4 


.08866 


11.2789 


.10628 


9.40904 


.12397 


8.06674 


•14173 


7^05579 


.15958 


6.26655 


56 


5 


.08895 


II. 2417 


.10657 


9-38307 


.12426 


8.04756 


.14202 


7^04105 


.15988 


6.25486 


55 


6 


.08925 


11.2048 


.10687 


9-35724 


.12456 


8.02848 


.14232 


7.02637 


.16017 


6.24321 


54 


7 


.08954 


11.1681 


.10716 


9-33155 


.12485 


8.00948 


.14262 


7.01174 


.16047 


6.23160 


S3 


8 


.08983 


11.1316 


.10746 


9-30599 


• 12515 


7.99058 


.14291 


6.99718 


.16077 


6.22003 


52 


9 


.09013 


11.0954 


•I077S 


9.28058 


.12544 


7-97176 


.14321 


6.98268 


.16107 


6.20851 


51 


10 


.09042 


11.0594 


.10805 


9-25530 


.12574 


7-95302 


.14351 


6.96823 


.16137 


6.19703 


SO 


11 


.09071 


11.0237 


.10834 


9.23016 


.12603 


7.93438 


.14381 


6.9538s 


.\(^\(>'j 


6.18559 


49 


12 


.09101 


10.9882 


.10863 


9.20516 


.12633 


7-91582 


.14410 


6.93952 


.16196 


6.17419 


48 


13 


.09130 


10.9529 


.10893 


9.18028 


.12662 


7-89734 


.14440 


6.92525 


.16226 


6.16283 


47 


14 


.09159 


10.9178 


.10922 


9- 1 5554 


.12692 


7-87895 


.14470 


6.91104 


.16256 


6.15151 


46 


IS 


.09189 


10.8829 


.10952 


9-13093 


.12722 


7.86064 


.14499 


6.89688 


.16286 


6.14023 


45 


i6 


.09218 


10.8483 


.10981 


9.10646 


.12751 


7-84242 


.14529 


6.88278 


.16316 


6.12899 


44 


17 


.09247 


10.8139 


.IIOII 


9.08211 


.12781 


7-82428 


.14559 


6.86874 


.16346 


6.11779 


43 


i8 


•09277 


10.7797 


.11040 


9-05789 


.12810 


7-80622 


.14588 


6.85475 


.16376 


6.10664 


42 


19 


.09306 


10.7457 


.11070 


9-03379 


.12840 


7-78825 


.14618 


6.84082 


.16405 


6.09552 


41 


20 


.09335 


10.7119 


.11099 


9.00983 


.12869 


7.7703s 


.14648 


6.82694 


.16435 


6.08444 


40 


21 


.09365 


10.6783 


.11128 


8.98598 


.12899 


7.75254 


.14678 


6.81312 


.16465 


6.07340 


39 


22 


.09394 


10.6450 


.11158 


8.96227 


.12929 


7.73480 


.14707 


6.79936 


.16495 


6.06240 


38 


23 


.09423 


10.6118 


.11187 


8.93867 


.12958 


7.7171S 


.14737 


6.78564 


.16525 


6.05143 


37 


24 


.09453 


10.5789 


.11217 


8.91520 




7-69957 


.14767 


6.77199 


.16555 


6.04051 


36 


25 


.09482 


10.5462 


.11246 


8.89185 


.13017 


7.68208 


.14796 


6.75838 


.16585 


6.02962 


35 


26 


.09511 


10.5136 


.11276 


8.86862 


.13047 


7-66466 


.14826 


6.74483 


.16615 


6.01878 


34 


27 


.09541 


10.4813 


.11305 


8.84551 


.13076 


7-64732 


.14856 


6.73133 


.16645 


6.00797 


33 


28 


.09570 


10.4491 


.11335 


8.82252 


.13106 


7-63005 


.14886 


6.71789 


.16674 


5.99720 


Z2 


29 


.09600 


10.4172 


.11364 


8.79964 


.13136 


7.61287 


.14915 


6.70450 


.16704 


5 -98646 


31 


30 


.09629 


10.3854 


.11394 


8.77689 


.13165 


7-59575 


.14945 


6.69116 


.16734 


5.97576 


30 


31 


.09658 


10.3538 


.11423 


8.75425 


.13195 


7.57872 


.14975 


6.67787 


.16764 


5^96510 


29 


32 


.09688 


10.3224 


.11452 


8.73172 


.13224 


7-56176 


.15005 


6.66463 


.16794 


S.95448 


28 


33 


.09717 


10.2913 


.11482 


8.70931 


.13254 


7-54487 


.15034 


6.65144 


.16824 


5^94390 


27 


34 


.09746 


10.2602 


.iiSii 


8.68701 


.13284 


7.52806 


.15064 


6.63831 


.16854 


5^93335 


26 


35 


.09776 


10.2294 


-IIS4I 


8.66482 


.13313 


7-51132 


.15094 


6.62523 


.16884 


5-92283 


25 


36 


.09805 


10.1988 


.11570 


8.6427s 


.13343 


7-49465 


.15124 


6 61219 


.16914 


5-91236 


24 


37 


.09834 


10.1683 


.11600 


8.62078 


•13372 


7.47806 


-15153 


6.59921 


.16944 


5.90191 


23 


38 


.09864 


10.1381 


.11629 


8.59893 


.13402 


7.46154 


.15183 


6.58627 


.16974 




22 


39 


.09893 


10.1080 


.11659 


8.57718 


•13432 


7-44509 


• 15213 


6.57339 


.17004 


5 88114 


21 


40 


.09923 


10.0780 


.11688 


8.55555 


.13461 


7.42871 


.15243 


6.560SS 


.17033 


5-87080 


20 


41 


.09952 


10.0483 


.11718 


8.53402 


.13491 


7-41240 


.15272 


6.54777 


.17063 


5 -8605 1 


19 


42 


.09981 


10.0187 


.11747 


8.51259 


.13521 


7-39616 


.15302 


6.53503 


.17093 


5-85024 


18 


43 


.10011 


9.98931 


.11777 


8.49128 


• 13550 


7.37999 


.15332 


6.52234 


.17123 


5.84001 


17 


44 


.10040 


9.96007 


.11806 


8.47007 


.13580 


7-36389 


• 15362 


6.50970 


.17153 


5.82982 


16 


45 


.10069 


9-93101 


.11836 


8.44896 


.13609 


7-34786 


• 15391 


6.49710 


.17183 


5.81966 


15 


46 


.10099 


9.90211 


.11865 


8.42795 


• 13639 


7-33190 


.15421 


6.48456 


.17213 


5-80953 


14 


47 


.10128 


9-87338 


.11895 


8.40705 


.13669 


7.31600 


.15451 


6.47206 


.17243 


5-79944 


13 


48 


.10158 


9-84482 


.11924 


8.3862s 


.13698 


7.30018 


.15481 


6.45961 


.17273 


S-78938 


12 


49 


.10187 


9.81641 


.11954 


8.36555 


.13728 


7.28442 


•ISSII 


6.44720 


.17303 


S.77936 


11 


50 


.10216 


9.78817 


.11983 


8.34496 


.13758 


7.26873 


.15540 


6.43484 


.-i-llU 


5-76937 


10 


SI 


.10246 


9-76009 


.12013 


8.32446 


.13787 


7-25310 


.15570 


6.42253 


.17363 


5-75941 


9 


52 


.10275 


9-73217 


.12042 


8.30406 


.13817 


7-23754 


.15600 


6.41026 


.17393 


5-74949 


8 


S3 


.10305 


9-70441 


.12072 


8.28376 


.13846 


7.22204 


.15630 


6.39804 


.17423 


5-73960 


7 


54 


.10334 


9-67680 


.12101 


8.26355 


.13876 


7.20661 


.15660 


6.38587 


.17453 


5-72974 


6 


55 


.10363 


9.64935 


.12131 


8.24345 


.13906 


7-19125 


.15689 


6-37374 


.17483 


5-71992 


S 


S6 


.10393 


9.62205 


.12160 


8.22344 


.13935 


7-17594 


.15719 


6.36165 


.17513 


5-7IOI3 


4 


S7 


.10422 


9.59490 


.12190 


8.20352 


.13965 


7-16071 


• 15749 


6.34961 


• 17543 


5-70037 


3 


58 


.10452 


9.56791 


.12219 


8.18370 


.13995 


7-I45S3 


.15779 


6.33761 


•17573 


5-69064 


2 


59 


.10481 


9-54106 


.12249 


8.16398 


.14024 


7-13042 


.15809 


6.32566 


.17603 


S-68094 


I 


60 


.10510 


9-51436 


.12278 


8.14435 


.14054 


7-II537 


.15838 


6.31375 


.17633 


5-67128 





/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


84° 


8: 


i° 


82° 


8] 





80° 



195 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTS AND COTANGENTS 



1 


10° 


11° 


12° 


13° 


14° 


/ 


Tangf 


Cotangr 


Tang 


Cotang: 


Tangf 


Cotangf 


Tangf 


Cotangf 


Tangf 


Cotangf 


o 


.17633 


S.67128 


.19438 


S-I445S 


.21256 


4.70463 


.23087 


4.33148 


-24933 


4.01078 


60 


I 


.17663 


5.66165 


.19468 


S-13658 


.21286 


4.69791 


.23117 


4-32573 


.24964 


4.00582 


59 


2 


.17693 


5.65205 


-19498 


5.12862 


.21316 


4.69121 


-23148 


4-32001 


-24995 


4.00086 


S8 


3 


.17723 


S.64248 


.19529 


5.12069 


.21347 


4.68452 


.23179 


4.31430 


.25026 


3-99592 


57 


4 


.17753 


5.63295 


-19559 


5.11279 


•21377 


4-67786 


.23209 


4.30860 


•25056 


3.99099 


56 


S 


.17783 


5.62344 


-19589 


5.10490 


.21408 


4.67121 


.23240 


4-30291 


.25087 


3.98607 


55 


6 


.17813 


5. 61397 


.19619 


5-09704 


.21438 


4.66458 


.23271 


4.29724 


.25118 


3.98117 


54 


7 


.17843 


S.60452 


.19649 


5-08921 


.21469 


4-65797 


.23301 


4.29159 


•25149 


3-97627 


53 


8 


.17873 


5-59511 


.19680 


5-08139 


.21499 


4-65138 


.23332 


4.28595 


.25180 


3-97139 


52 


9 


.17903 


5.58573 


.19710 


5-07360 


.21529 


4-64480 


-23363 


4.28032 


.25211 


3-96651 


51 


10 


.17933 


S.57638 


.19740 


5.06584 


.21560 


4.6382s 


• 222,92, 


4.27471 


•25242 


3-96165 


50 


II 


.17963 


5.56706 


.19770 


S-05809 


.21590 


4.63171 


.23424 


4.26911 


.25273 


3.95680 


49 


12 


• 17993 


5.55777 


.19801 


5-05037 


.21621 


4.62518 


.23455 


4.26352 


•25304 


3-95196 


48 


13 


.18023 


5.54851 


.19831 


5-04267 


.21651 


4.61868 


.23485 


4.25795 


-25335 


3-94713 


47 


14 


•'^°^^ 


5.53927 


.19861 


5-03499 


.21682 


4.61219 


.23516 


4.25239 


.25366 


3-94232 


46 


IS 


.18083 


5.53007 


.19891 


5-02734 


.21712 


4.60572 


•23547 


4.24685 


-25397 


3-93751 


45 


i6 


.18113 


5.52090 


.19921 


5.01971 


.21743 


4-59927 


•23578 


4-24132 


.25428 


3-93271 


44 


17 


.18143 


5-51176 


.19952 


5.01210 


.21773 


4.59283 


.23608 


4-23580 


.25459 


3-92793 


<3 


i8 


.18173 


5.50264 


.19982 


5-00451 


.21804 


4-58641 


.23639 


4-23030 


.25490 


3.92316 


42 


19 


.18203 


5-49356 


.20012 


4.99695 


.21834 


4.58001 


.23670 


4.22481 


-25521 


3-91839 


41 


20 


.18233 


5.48451 


.20042 


4-98940 


.21864 


4-57363 


.23700 


4.21933 


-25552 


3-91364 


40 


21 


.18263 


5-47548 


-20073 


4.98188 


.21895 


4-56726 


.2212\ 


4-21387 


.25583 


3-90890 


39 


22 


.18293 


5-46648 


.20103 


4-97438 


.21925 


4-56091 


.23762 


4.20842 


.25614 


3-90417 


. 38 


23 


.18323 


5-45751 


.20133 


4-96690 


.21956 


4-55458 


•23793 


4.20298 


.25645 


3-89945 


37 


24 


.18353 


5.44857 


.20164 


4-95945 


.21986 


4.54826 


•23823 


4-19756 


.25676 


3.89474 


36 


25 


.18384 


5. 43966 


.20194 


4-95201 


.22017 


4-54196 


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4-19215 


-25707 


3-89004 


35 


26 


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5.43077 


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4.94460 


.22047 


4-53568 


.23885 


4-18675 


.25738 


3-88536 


34 


27 , 


.18444 


5-42192 


.20254 


4.93721 


.22078 


4-52941 


.23916 


4-18137 


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3.88068 


33 


28 


.18474 


5-41309 


.20285 


4.92984 


.22108 


4-52316 


.23946 


4-17600 


.25800 


3-87601 


32 


29 


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5-40429 


.2031S 


4-92249 


.22139 


4-51693 


•23977 


4-17064 


-25831 


3.87136 


31 


30 


.18534 


S-39552 


-20345 


4-91516 


.22169 


4.51071 


.24008 


4-16530 


.25862 


3.86671 


30 


31 


.18564 


5-38677 


.20376 


4-90785 


.22200 


4-50451 


.24039 


4-15997 


.25893 


3.86208 


29 


32 


.18594 


S-37805 


.20406 


4-90056 


.22231 


4-49832 


.24069 


4-15465 


.25924 


3-85745 


28 


33 


..18624 


5-36936 


.20436 


4-89330 


.22261 


4-49215 


.24100 


4-14934 


.25955 


3-85284 


21 


34 


.18654 


5-36070 


.20466 


4.88605 


.22292 


4.48600 


.24131 


4-14405 


.25986 


3.84824 


■ 26 


35 


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S-35206 


.20497 


4-07882 


.22322 


4.47986 


.24162 


4-13877 


.26017 


3-84364 


25 


36 


.18714 


5.34345 


-20527 


4-87162 


.22353 


4.47374 


•24193 


4-13350 


.26048 


3-83906 


24 


37 


.18745 


S-33487 


.20557 


4-86444 


.22383 


4-46764 


•24223 


4-12825 


.26079 


3-83449 


22 


38 


.18775 


S-32631 


.20588 


4-85727 


.22414 


4-46155 


-24254 


4.12301 


.26110 


3-82992 


22 


39 


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5-31778 


.20618 


4-85013 


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4-45548 


-24285 


4-11778 


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3.82537 


21 


40 


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5-30928 


.20648 


4-84300 


-22475 


4-44942 


-24316 


4.II256 


.26172 


3-82083 


20 


41 


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5-30080 


.20679 


4-83590 


.22505 


4-44338 


.24347 


4-10736 


.26203 


3-81630 


19 


42 


.18895 


5-29235 


.20709 


4.82882 


.22536 


4-43735 


.24377 


4.10216 


.26235 


3.81177 


18 


43 


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5-28393 


.20739 


4.82175 


.22567 


4.43134 


.24408 


4.09699- 


.26266 


3.80726 


17 


44 


.18955 


5-27553 


.20770 


4-81471 


.22597 


4-42534 


.24439 


4.09182 


.26297 


3.80276 


16 


45 


.18986 


S-26715 


.20800 


4.80769 


.22628 


4.41936 


.24470 


4.08666 


.26328 


3-79827 


15 


46 


.19016 


5-25880 


.20830 


4.80068 


.22658 


4-41340 


-24501 


4.08152 


.26359 


3.79378 


14 


47 


.19046 


5-25048 


.20861 


4-79370 


.22689 


•4-40745 


.24532 


4.07639 


.26390 


3-78931 


13 


48 


.19076 


5-24218 


.20891 


4-78673 


.22719 


4.40152 


.24562 


4-07127 


.26421 


3-78485 


12 


49 


.19106 


5-23391 


.20921 


4.77978 


.22750 


4.39560 


-24593 


4.06616 


.26452 


3.78040 


II 


SO 


.19136 


5-22566 


.20952 


4-77286 


.22781 


4.38969 


.24624 


4.06107 


.26483 


3.77595 


10 


51 


.19166 


5-21744 


.20982 


4.76595 


.22811 


4.38381 


-24655 


4.05599 


.26515 


3.77152 


9 


52 


.19197 


5-20925 


.21013 


4-75906 


.22842 


4.37793 


.24686 


4.05092 


.26546 


3-76709 


8 


S3 


.19227 


5. 20107 


.21043 


4.75219 


.22872 


4.37207 


.24717 


4.04586 


.26577 


3.76268 


7 


54 


•19257 


5-19293 


.21073 


4-74534 


.22903 


4.36623 


-24747 


4.04081 


.26608 


3.75828 


6 


55 


.19287 


5.18480 


.21104 


4-73851 


.22934 


4.36040 


.24778 


4.03578 


.26639 


3.75388 


5 


56 


.19317 


5. 17671 


.21134 


4-73170 


.22964 


4.35459 


.24809 


4 03076 


.26670 


3-74950 


4 


57 


.19347 


5.16863 


.21164 


4.72490 


.22995 


4.34879 


.24840 


4.02574 


.26701 


3-74512 


2 


58 


.19378 


5-16058 


.21195 


4.71813 


.23026 


4.34300 


.24871 


4-02074 


-26733 


3-74075 


2 


59 


.19408 


5-15256 


.21225 


4-71137 


.23056 


4-33723 


.24902 


4-01576 


.26764 


3-73640 


I 


60 


.19438 


5-14455 


.21256 


4.70463 


.23087 


4.33148 


-24933 


4.01078 


.26795 


3-7320S 





/ 


Cotang 


Tang 


Cotangf 


Tangf 


Cotangf 


Tang- 


Cotangf 


Tangf 


Cotangf 


Tangf 


/ 


79° ' 


78° 


77° 


76° 


7 





196 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTS AND COTANGENTS 



1 


15° 


16° 


17° 


18° 


K 


P° 


/ 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


Cotang: 


Tang: 


Cotansr 


Tang: 


Cotang: 





.2679s 


3-73205 


.28675 


3.48741 


.30573 


3-27085 


.32492 


3.07768 


.34433 


2.90421 


60 


I 


.26826 


3.72771 


.28706 


3.48359 


.30605 


3-26745 


.32524 


3.07464 


.34465 


2.90147 


59 


2 


.26857 


3.72338 


.28738 


3-47977 


.30637 


3.26406 


.32556 


3.07160 


.34498 


2.89873 


58 


3 


.26888 


3.71907 


.28769 


3-47596 


.30669 


3.26067 


.32588 


3.06857 


.34530 


2.89600 


57 


4 


.26920 


3.71476 


.28800 


3-47216 


.30700 


3.25729 


.32621 


3.06554 


.34563 


2.89327 


56 


S 


.26951 


3.71046 


.28832 


C.46837 


.30732 


3.25392 


.32653 


3.06252 


-34596 


2.890SS 


55 


6 


.26982 


3.70616 


.28864 


3.46458 


.30764 


3.25055 


.3268s 


3-05950 


.34628 


2.88783 


54 


7 


.27013 


3.70188 


.28895 


3.46080 


.30796 


3.24719 


.32717 


3.05649 


.34661 


2.88511 


S3 


8 


.27044 


3.69761 


.28927 


3-45703 


.30828 


3.24383 


.32749 


3-05349 


.34693 


2.88240 


52 


9 


.27076 


3.69335 


.28958 


3-45327 


.30860 


3.24049 


.32782 


3.OS049 


.34726 


2.87970 


SI 


10 


.27107 


3.68909 


.28990 


3.44951 


.30891 


3.23714 


.32814 


3.04749 


.34758 


2.87700 


50 


II 


.27138 


3.6848s 


.29021 


3.44576 


.30923 


3.23381 


.32846 


3-04450 


.34791 


2.87430 


49 


12 


.27169 


3.68061 


.29053 


3.44202 


•30955 


3.23048 


.32878 


3-04152 


.34824 


2.87161 


48 


13 


.27201 


3.67638 


.29084 


3-43829 


.30987 


3.22715 


.32911 


3-03854 


.34856 


2.86892 


47 


14 


,27232 


3.67217 


.29116 


3-43456 


.31019 


3.22384 


.32943 


3.03556 


.34889 


2.86624 


46 


IS 


.2726J 


3.66796 


.29147 


3.43084 


.31051 


3.22053 


.32975 


3.03260 


.34922 


2.86356 


45 


i4 


.27294 


3.66376 


.29179 


3.42713 


.31083 


3.21722 


.33007 


3-02963 


.34954 


2.86089 


44 


I? 


.27326 


3.65957 


.29210 


3.42343 


.3111S 


3.21392 


.33040 


3-02667 


.34987 


2.85822 


43 


i8 


.27357 


3.65538 


.29242 


3-41973 


.31147 


3-21063 


.33072 


3-02372 


.35020 


2-85555 


42 


19 


.27388 


3.65121 


.29274 


3.41604 


.31178 


3-20734 


.33104 


3-02077 


.35052 


2.85289 


41 


20 


.27419 


3.6470s 


.2930s 


3-41236 


.31210 


3.20406 


.33136 


3.01783 


.35085 


2.85023 


40 


21 


.274SI 


3.64289 


.29337 


3.40869 


.31242 


3-20079 


.33169 


3.01489 


.35118 


2.84758 


39 


22 


.274S2 


3.63874 


.20368 


3.40502 


.31274 


3-19752 


.33201 


3.01196 


.35150 


2.84494 


38 


23 


.27513 


3.63461 


.29400 


3.40136 


.31306 


3-19426 


-33233 


3-00903 


.35183 


2.84229 


37 


24 


.27545 


3.63048 


.29432 


3.39771 


.31338 


3.19100 


.33266 


3.00611 


.35216 


2.8396s 


36 


25 


.27576 


3.62636 


.29463 


3.39406 


.31370 


3.18775 


.33298 


3.00319 


.35248 


2.83702 


35 


26 


.27607 


3.62224 


.29495 


3.39042 


.31402 


3-18451 


.33320 


3.00028 


.35281 


2-83439 


34 


27 


.27633 


3.61814 


.29526 


3.38679 


.31434 


3.18127 


.23363 


2.99738 


.35314 


2.83176 


33 


28 


.27670 


3.614OS 


.29553 


3.38317 


.31466 


3-17804 


.33395 


2.99447 


.35346 


2.82914 


32 


29 


.27701 


3.60996 


.295QO 


3.37955 


.31498 


3-17481 


.33427 


2.99158 


.35379 


2.82653 


31 


30 


.27732 


3.60588 


.29621 


3-37594 


.31530 


3.17159 


.33460 


2.98868 


.35412 


2.82391 


30 


31 


.27764 


3.60181 


.29653 


3-37234 


.31562 


3.16838 


.33492 


2.98580 


.3S44S 


2.82130 


29 


32 


.27795 


3.59775 


.29685 


3-36875 


-31594 


3-16517 


.33524 


2.98292 


.35477 


2.81870 


28 


33 


.27826 


3.59370 


.29716 


3-36516 


.31626 


3-16197 


.33557 


2.98004 


•35510 


2.81610 


27 


34 


.27858 


3.53966 


.23748 


3.36158 


.31658 


3.15877 


.33589 


2.97717 


.35543 


2.81350 


26 


35 


.27889 


3.53562 


.29780 


3.35800 


.31690 


3-15558 


.33621 


2.97430 


.35576 


2.81091 


25 


36 


.27921 


3.53160 


.29811 


3-35443 


.31722 


3-15240 


.33654 


2.97144 


.35608 


2.80833 


24 


37 


.27952 


3.57758 


.29843 


3.35087 


.31754 


3.14922 


.33686 


2.96858 


•35641 


2.80574 


23 


38 


.279S3 


3.57357 


.29875 


3-34732 


.31786 


3.14605 


.33718 


2.96573 


.35674 


2.80316 


22 


39 


.28015 


3.56957 


.29906 


3-34377 


.31818 


3.14288 


.33751 


2.962S3 


.35707 


2.80059 


21 


40 


.28046 


3.56557 


.29938 


3.34023 


.31850 


3.13972 


.33783 


2.96004 


.35740 


2;798o2 


20 


41 


.28077 


3.56159 


.29970 


3-33670 


.31882 


3.13656 


.33816 


2.9S72I 


.35772 


2.79545 


19 


42 


.28109 


3.55761 


.30001 


3-333-^7 


.31914 


3.13341 


.33848 


2.95437 


.3580s 


2.79289 


18 


43 


.28140 


3.55364 


.30033 


3.3296s 


.31946 


3-13027 


.33881 


2.9515s 


.35838 


2.79033 


17 


44 


.28172 


3.54968 


.30065 


3.32614 


•31978 


3.12713 


.33913 


2.94872 


.35871 


2.78778 


16 


45 


.28203 


3-54573 


.30097 


3.32264 


.32010 


3.12400 


.33945 


2.94591 


.35904 


2.78523 


15 


46 


.28234 


3-54179 


.30128 


3.31914 


.32042 


3.12087 


.33978 


2.94309 


.35937 


2.78269 


14 


47 


.28266 


3-53785 


.30160 


3.31565 


.32074 


3-II77S 


.34010 


2.94028 


.35969 


2.78014 


13 


48 


.28297 


3-53353 


.30192 


3.31216 


.32106 


3.11464 


.34043 


2.93743 


.36002 


2.77761 


12 


49 


.28329 


3.53001 


.30224 


3.30868 


.32139 


3-I11S3 


.34075 


2.93468 


.360,3s 


2.77507 


II 


SO 


.28360 


3.52609 


.30255 


3-30521 


•32171 


3.10842 


.34108 


2.93189 


.36068 


2.77254 


10 


SI 


.28391 


3.52219 


.30287 


3.30174 


.32203 


3-10532 


.34140 


2.92910 


.36101 


2.77002 


9 


52 


.28423 


3.51829 


.30319 


3.29829 


.32235 


3.10223 


.34173 


2.92632 


.36134 


2.76750 


8 


S3 


.28454 


3.51441 


.30351 


3-29483 


.3^267 


3.09914 


.34205 


2.92354 


.36167 


2.76498 


7 


54 


.28486 


3.51053 


.30382 


3.29139 


•32299 


3.09606 


.34238 


2.92076 


.36199 


2.76247 


6 


5S 


.28517 


3.50666 


.30414 


3.28795 


.32331 


3.09298 


.34270 


2.91799 


.36232 


2.75996 


S 


56 


.28549 


3.50279 


.30446 


3.28452 


.32363 


S.08991 


.34303 


2.9i5.-:3 


.3626s 


2.75746 


4 


57 


.28580 


3.49894 


.30478 


3.28109 


.32396 


3.08685 


.34335 


2.91246 


.36298 


2.75496 


3 


58 


.28612 


3.49509 


.30509 


3.27767 


.32428 


3.08379 


.34368 


2.90971 


.36331 


2.75246 


2 


59 


.28643 


3-49T2S 


.30541 


3.27426 


.32460 


3.08073 


.34400 


2.90656 


.36364 


2.74997 


I 


6o 


.28675 


3.48741 


.30573 


3.27085 


.32492 


3.07768 


-34433 


2.90421 


.363Q7 


2.;'4748 





/ 


Cotang 


Tang 


Cotang: 


Tang: 


CotanK 


Tan^ 


Cotang: 


Tangr 


Cotang: 


Tang: 




74° 


73° 


72° 


71° 


7< 


)° 



197 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTS AND COTANGENTS 



/ 


20^ 


21° 


22° 


23° 


24° 


/ 


Tang- 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


.36397 


2.74748 


.38386 


2.60509 


.40403 


2.47509 


.42447 


2.35585 


.44523 


2.24604 


60 


I 


.36430 


2.74499 


.3^420 


2.60283 


.40436 


2.47302 


.42482 


2.35395 


.44558 


2.^44^ 


59 


2 


.36463 


2.74251 


•3f4S3 


2.60057 


.40470 


2.4709s 


.42516 


2.35-205^ 


.44593 


2.24252 


58 ^ 


3 


.36496 


2.74004 


.38487 


2.59831 


.40504 


2.46888 


.42551 


^.35015 


.44627 


2.24077 


57 


4 


.36529 


2.73756 


.38520 


2.59606 


.40538 


2.46682 


.4258^5 


-2,3482s 


.44662 


2.23902 


56 


5 


.36562 


2.73509 


.38553 


2.59381 


.40572 


2.46475 


.42619 


2.34636 


.44697 


2.23727 


55 


6 


.36595 


2.73263 


.38587 


2.59156 


.40606 


2.46270 


.42654 


2.34447 


.44732 


2.23553 


54 


7 


.36628 


2.73017 


,38620 


2.58932 


.40640 


2.4606s 


.42688 


2.34258 


.44767 


2.23378 


S3 


8 


.36661 


2.72771 


.38654 


2.58708 


.40674 


2.45860 


.42722 


2.34069 


.44802 


2.23204 


52 


9 


.36694 


2.72526 


.38687 


2.58484 


.40707 


2.45655 


.42757 


2.33881 


.44837 


2.23030 


51 


10 


.36727 


2.72281 


.38721 


2.58261 


.40741 


2.45451 


.42791 


2.33693 


.44872 


2.22857 


SO 


II 


.36760 


2.72036 


.38754 


2.58038 


.40775 


2.45246 


.42826 


2.3350s 


.44907 


2.22683 


49 


12 


.36793 


2.71792 


.38787 


2.57815 


.40809 


2.45043 


.42860 


2.33317 


.44942 


2.22510 


48 


13 


.36826 


2.71548 


.38821 


2.57593 


.40843 


2.44839 


.42894 


2.33130 


.44977 


2.22337 


47 


14 


.36859 


2.71305 


.38854 


2.57371 


.40877 


2.44636 


.42929 


2.32943 


.45012 


2.22164 


46 


IS 


.36892 


2.71062 


.38888 


2.57150 


.40911 


2.44433 


.42963 


2.32756 


.45047 


2.21992 


45 


i6 


.36925 


2.70819 


.38921 


2.56928 


.40945 


2.44230 


.42998 


2.32570 


.45082 


2.21819 


44 


17 


.36958 


2.70577 


.38955 


2.56707 


.40979 


2.44027 


.43032 


2.32383 


.45117 


2.21647 


43 


i8 


.36991 


2.70335 


.38988 


2.56487 


.41013 


2.4382s 


.43067 


2.32197 


.45152 


2.21475 


42 


19 


.37024 


2.70094 


.39022 


2.56266 


.41047 


2.43623 


.43101 


2.32012 


.45187 


2.21304 


41 


20 


.37057 


2.69853 


.39055 


2.56046 


.41081 


2.43422 


.43136 


2.31826 


.45222 


2.21132 


40 


21 


.37090 


2.69612 


.39089 


2.55827 


.41115 


2.43220 


.43170 


2.31641 


.45257 


2.20961 


39 


22 


.37123 


2.69371 


.39122 


2.55608 


.41149 


2.43019 


.43205 


2.31456 


.45292 


2.20790 


38 


23 


.37157 


2.69131 


.39156 


2.55389 


.41183 


2.42819 


.43230 


2.31271 


.45327 


2.20619 


27 


24 


.37190 


2.68892 


.39190 


2.55170 


.41217 


2.42618 


.43274 


2.31086 


.45362 


2.20449 


36 


25 


.37223 


2.68653 


.39223 


2.54952 


.41251 


2.42418 


.43308 


2.30902 


.45397 


2.20278 


35 


26 


.37256 


2.68414 


.39257 


2.54734 


.41285 


2.42218 


.43343 


2.30718 


.45432 


2.20108 


34 


27 


.37289 


2.68175 


.39290 


2.54516 


.41319 


2.42019 


.43378 


2.30534 


.45467 


2.19938 


33 


28 


.37322 


2.67937 


.39324 


2.54299 


.41353 


2.41819 


.43412 


2.30351 


.45502 


2.19769 


32 


29 


.37355 


2.67700 


.39357 


2.54082 


.41387 


2.41620 


.43447 


2.30167 


.45538 


2.19599 


31 


30 


.37388 


2.67462 


.39391 


2.5386s 


.41421 


2.41421 


.43481 


2.29984 


.45573 


2.19430 


30 


31 


.37422 


2.67225 


.39425 


2.53648 


.41455 


2.41223 


.43516 


2.29801 


.45608 


2.19261 


29 


32 


.37455 


2.66989 


.39458 


2.53432 


.41490 


2.41025 


.43550 


2.29619 


.45643 


2.19092 


28 


33 


.37488 


2.66752 


.39492 


2.53217 


.41524 


2.40827 


.43585 


2.29437 


.45678 


2.18923 


27 


34 


.37521 


2.66516 


.39526 


2.53001 


.41558 


2.^0629 


.43620 


2.29254 


.45713 


2.18755 


26 


35 


.37554 


2.66281 


• 39559 


2.52786 


.41532 


2.40432 


.43654 


2.29073 


.45748 


2.18587 


25 


36 


.37588 


2.66046 


.39593 


2.52571 


.41626 


2.4023s 


.43689 


2.2S891 


.45784 


2.18419 


24 


37 


.37621 


2. 65811 


.39626 


2.52357 


.41660 


2.40038 


.43724 


2.28710 


.45819 


2.18251 


23 


38 


.37654 


2.65576 


.39660 


2.52142 


.41694 


2.39841 


.43758 


2.28528 


.45854 


2.18084 


22 


39 


.37687 


2.65342 


.39694 


2.51929 


.41728 


2.39645 


.43793 


2.28348 


.45889 


2.17916 


.21 


40 


.37720 


2.65109 


.39727 


2.51715 


.41763 


2.39449 


.43828 


2.28167 


.45924 


2.17749 


20 


41 


.37754 


2.64875 


.39761 


2.51502 


.41797 


2.39253 


.43862 


2.27987 


.45960 


2.17582 


19 


42 


.37787 


2.64642 


.39795 


2.51289 


.41831 


2.39058 


.43897 


2.27806 


.45995 


2.17416 


18 


43 


.37820 


2.64410 


.39829 


2.51076 


.41865 


2.38863 


.43932 


2.27626 


.46030 


2.17249 


17 


44 


.37853 


2.64177 


.39862 


2.50864 


.41899 


2.38668 


.43966 


2.27447 


.46065 


2.17083 


16 


45 


.37887 


2.6394s 


.39896 


2.50652 


.41933 


2.38473 


.44001 


2.27267 


.46101 


2.16917 


15 


46 


.37920 


2.63714 


.39930 


2.50440 


.41968 


2.38279 


.44036 


2.27088 


.46136 


2.16751 


14 


47 


.37953 


2.63483 


.39963 


2.50229 


.42002 


2.38084 


.44071 


2.26909 


.46171 


2.1658s 


13 


48 


.37986 


2.63252 


.39997 


2.50018 


.42036 


2.37891 


.44105 


2.26730 


.46206 


2.16420 


12 


49 


.38020 


2.63021 


.40031 


2.49807 


.42070 


2.37697 


.44140 


2.26552 


.46242 


2.16255 


II 


SO 


.38053 


2.62791 


.40065 


2.49597 


.42105 


2.37504 


.44175 


2.26374 


.46277 


2.16090 


10 


SI 


.38086 


2.62561 


.40098 


2.49386 


.42139 


2.37311 


.44210 


2.26196 


.46312 


2.15925 


9 


52 


.38120 


2.62332 


.40132 


2.49177 


.42173 


2.37118 


.44244 


2.26018 


.46348 


2.15760 


8 


53 


.38153 


2.62103 


.40166 


2.48967 


.42207 


2.3692s 


.44279 


2.25840 


.46383 


2.15596 


7 


54 


.38186 


2.61874 


.40200 


2.48758 


.42242 


2.36733 


.44314 


2.25663 


.46418 


2.15432 


6 


55 


.38220 


2.61646 


.40234 


2.48549 


.42276 


2.36541 


.44349 


2.25486 


.46454 


2.15268 


s 


S6 


.38253 


2.61418 


.40267 


2.48340 


.42310 


2.36349 


.44384 


2.25309 


.46489 


2.15104 


4 


57 


.38386 


2.61190 


.40301 


2.48132 


.42345 


2.36158 


.44418 


2.25132 


.46525 


2.14940 


3 


58 


.38320 


2.60963 


.40335 


2.47924 


.42379 


2.35967 


.44453 


2.24956 


.46560 


2.14777 


2 


59 


.38353 


2.60736 


.40369 


2.47716 


.42413 


2.35776 


.44488 


2.24780 


.46595 


2.14614 


I 


60 


.38386 


2.60509 


.40403 


2.47509 


.42447 


2.35585 


.44523 


2.24604 


.46631 


2.I44SI 





/ 


Cotang 


Tang- 


Cotang 


Tang 


Cotang 


Tangr 


Cotang 


Tang 


Cotang 


Tang 


/ 


6c 


f 


68° 


67° 


66° 


65° 



198 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTvS AND COTANGENTS 



1 


25° 


26° 


27° 


28° 


29° 


/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.46631 


2.14451 


.48773 


2.05030 


.50953 


1.96261 


.53171 


1.88073 


.55431 


1.80405 


60 




.46666 


2.14288 


.48809 


2.04879 


.50989 


1.96120 


.53208 


1.87941 


• 55469 


1.80281 


59 


2 


.46702 


2.1412s 


.48845 


2.04728 


.51026 


1^95979 


.53246 


1.87809 


■ 55507 


1.80158 


58 


3 


.46737 


2.13963 


.48881 


2.04577 


.51063 


1^95838 


.53283 


1.87677 


•55545 


1.80034 


57 


4 


.46772 


2.13801 


.48917 


2.04426 


.51099 


1.95698 


.53320 


1.87546 


•55583 


1.79911 


S6 


s 


.46808 


2.13639 


.48953 


2.04276 


.51136 


1^95557 


.53358 


1.87415 


• 55621 


1.79788 


55 


6 


.46843 


2.134-7 


.48989 


2.04125 


.51173 


1^95417 


.53395 


1.87283 


•55659 


1.79665 


54 


7 


.46879 


2.13316 


.49026 


2.03975 


.51209 


1^95277 


.53432 


1.87152 


.55697 


1.79542 


53 


8 


.46914 


2.13154 


.49062 


2.0382s 


.51246 


1^95137 


.53470 


1.87021 


.55736 


1.79419 


52 


9 


.46950 


2.12993 


.49098 


2.03675 


.51283 


1.94997 


.53507 


1.86891 


.55774 


1.79296 


51 


10 


.46985 


2.12832 


•49134 


2.03526 


.51319 


1^94858 


.53545 


1.86760 


.55812 


1.79174 


50 


II 


.47021 


2.12671 


•49170 


2.03376 


•S1356 


1^94718 


.53582 


1.86630 


.55850 


1. 79051 


49 


12 


.47056 


2.12511 


.49206 


2.03227 


.51393 


1^94579 


.53620 


1.86499 


.55888 


1.78929 


48 


13 


.47092 


2.12350 


.49242 


2.03078 


.51430 


1.94440 


.53657 


1 86369 


.55926 


1.78807 


47 


14 


.47128 


2.12190 


.49278 


2.02929 


.51467 


1.94301 


.53694 


1.86239 


.55964 


1.78685 


46 


IS 


.47163 


2.12030 


.49315 


2.02780 


.51503 


1.94162 


•53732 


1.86109 


.56003 


1.78563 


45 


i6 


.47199 


2.11871 


.49351 


2.02631 


.51540 


1.94023 


■53769 


1-85979 


•56041 


1.78441 


•44 


17 


.47234 


2.11711 


.49387 


2.02483 


.51577 


1.93885 


•53807 


1.85850 


•56079 


1.78319 


43 


i8 


.47270 


2.11552 


.49423 


2.02335 


.51614 


1.93746 


• 53844 


1.85720 


•56117 


1.78198 


42 


19 


.4730s 


2.11392 


.49459 


2.02187 


.51651 


1.93608 


.53882 


1.85591 


•56156 


1.78077 


41 


20 


.47341 


2.11233 


.49495 


2.02039 


.51688 


1.93470 


.53920 


1.85462 


•56194 


1.77955 


40 


21 


.47377 


2.11075 


.49532 


2.01891 


.51724 


1.93332 


.53957 


1.85333 


•56232 


1.77834 


39 


22 


.47412 


2.10916 


.49568 


2.01743 


.51761 


I.93195 


.53995 


1.85204 


.56270 


1.77713 


•38 


2i 


.47448 


2.10758 


.49604 


2.01596 


.51798 


1.93057 


.54032 


1.85075 


•56309 


1.77592 


37 


24 


.47483 


2.10600 


.49640 


2.01449 


.51835 


1.92920 


.54070 


1.84946 


• 56347 


1.77471 


36 


25 


.47519 


2.10442 


.49677 


2.01302 


.51872 


1.92782 


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1.84818 


•56385 


1.77351 


35 


26 


.47555 


2.10284 


.49713 


2.01155 


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1.92645 


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1.84689 


.56424 


I 11210 


34 


27 


.47590 


2.10126 


.49749 


2.01008 


.51946 


1.92508 


.54183 


1.84561 


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I.771IO 


33 


28 


.47626 


2.09969 


.49786 


2.00862 


.51983 


1. 92371 


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1.84433 


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1.76990 


32 


29 


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2.09811 


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2.00715 


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1.92235 


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1.84305 


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1.76869 


31 


30 


.47698 


2.09654 


.49858 


2.00569 


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1.92098 


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1.84177 


.56577 


1.76749 


30 


31 


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2.09498 


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2.00423 


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I.91962 


.54333 


1.84049 


.56616 


1.76629 


29 


Z2 


.47769 


2.09341 


.49931 


2.00277 


.52131 


1. 91826 


.54371 


1.83922 


.56654 


1.76510 


28 


33 


.47805 


2.09184 


.49967 


2.00131 


.52168 


1. 91690 


.54409 


1.83794 


.56693 


1-76390 


27 


34 


.47840 


2.09028 


.50004 


1.99986 


.52205 


1. 91554 


.54446 


1.83667 


.56731 


1.76271 


26 


35 


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2.08872 


.50040 


1.99841 


.52242 


1.91418 


.54484 


1.83540 


.56769 


1.76151 


25 


36 


.47912 


2.08716 


.50076 


1.9969s 


-52279 


1.91282 


.54522 


1.83413 


.56808 


1-76032 


24 


37 


.47948 


2.08560 


.50113 


1-99550 


-52316 


1.91147 


.54560 


1.83286 


.56846 


1. 75913 


23 


38 


.47984 


2.08405 


.50149 


1.99406 


-52353 


I. 91012 


.54597 


1.83159 


.56885 


1.75794 


22 


39 


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2.08250 


.50185 


1.99261 


-52390 


1.90876 


.54635 


1.83033 


.56923 


1.75675 


21 


40 


.48055 


2.08094 


.50222 


1.99116 


.52427 


1.90741 


.54673 


1.82906 


.56962 


1.75556 


20 


41 


.48091 


2.07939 


.50258 


1.98972 


-52464 


1.90607 


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1.82780 


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1-75437 


19 


42 


.48127 


2.07785 


.50295 


1.98828 


.52501 


1.90472 


.54748 


1.82654 


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1-75319 


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2.07630 


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1.98684 


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1.90337 


.54786 


1.82528 


.57078 


1.75200 


17 


44 


.48198 


2.07476 


.50368 


1.98540 


.52575 


1.90203 


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1.82402 


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1.75082 


16 


45 


.48234 


2.07321 


.50404 


1.98396 


.52613 


1.90069 


.54862 


1.82276 


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1.74964 


15 


46 


.48270 


2.07167 


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1.98253 


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1.89935 


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1.82150 


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1.74846 


14 


47 


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2.07014 


•50477 


1.98110 


.52687 


1.89801 


.54938 


1.82025 


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1.74728 


13 


48 


.48342 


2.06860 


.50514 


1.97966 


•52724 


1.89667 


.54975 


1.81899 


.57271 


1.74610 


12 


49 


.48378 


2.06706 


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1.97823 


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1.89533 


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1-81774 


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1-74492 


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.48414 


2.06553 


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1.97681 


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1.89400 


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1.81649 


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1.74375 


10 


51 


.48450 


2.06400 


.50623 


1.97538 


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1.89266 


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1.81524 


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1.74257 


9 


52 


.48486 


2.06247 


.50660 


1-97395 


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1.89133 


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1.81399 


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I. 74140 


8 


53 


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2.06094 


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1-97253 


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1.89000 


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1.81274 


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1.74022 


7 


54 


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2.05942 


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1.97111 


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1.88867 


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1.81150 


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1.7390S 


6 


55 


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2.05790 


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1.96969 


-52985 


1.88734 


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1. 81025 


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1-73788 


5 


56 


.48629 


2.05637 


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1.96827 


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1.80901 


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1-73671 


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2.05485 


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1.80777 


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1-73555 


3 


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2.05333 


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1.96544 


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1.88337 


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1.80653 


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1-73438 


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2.05182 


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1.96402 


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1.88073 


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1-73205 





1 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


64° 


63° 


6i 


2° 


6 


[° 


6 


0° 



199 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTS AND COTANGENTS 



/ 


30° 


31 





32 





33° 


34° 


/ 


Tang: 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


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1.73205 


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60 


I 


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1.66318 


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1-59930 


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1.53888 


-67493 


1.48163 


59 


2 


.57813 


1.72973 


.60165 


1.66209 


.62568 


1.59826 


.65024 


1^53791 


-67536 


1.48070 


58 


3 


.57851 


1.72857 


.60205 


1.66099 


.62608 


1.59723 


.65065 


1^53693 


.67578 


1.47977 


57 


4 


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1. 72741 


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1.65990 


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1.59620 


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1^53595 


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1.4788s 


56 


S 


.57929 


1.7262s 


.60284 


1.65881 


.62689 


1.59517 


•65148 


1^53497 


.67663 


1.47792 


55 


6 


.57968 


i^72S09 


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1.65772 


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1. 59414 


•65189 


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1-47699 


54 


7 


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1.65663 


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1.72278 


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1. 65554 


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1.53205 


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1-47514 


52 


9 


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51 


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1.72047 


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1.59002 


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50 


II 


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1.65228 


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1.58900 


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1.47238 


49 


12 


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1.65120 


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1.52816 


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1.47146 


48 


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1.52719 


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47 


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46 


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44 


17 


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41 


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1.46320 


39 


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38 


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37 


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36 


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35 


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34 


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1. 70106 


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1^63505 


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1.57271 


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1.51370 


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33 


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32 


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30 


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29 


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1.62972 


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1.56767 


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1.50893 


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28 


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1.50797 


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1.62760 


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24 


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1.56265 


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23 


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20 


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IS 


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14 


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13 


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12 


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10 


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9 


52 


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8 


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1.60761 


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1-54675 


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1.48909 


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7 


54 


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1.60657 


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1-54576 


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6 


55 


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1.66978 


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1.60553 


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1.54478 


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1.48722 


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1-43258 


5 


56 


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1.66867 


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1.60241 


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2 


59 


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1.66538 


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1.60137 


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1-54085 


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I 


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1.48256 


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I.4281S 





/ 


Cotang- 


Tang- 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


59° 


5< 


3° 


5: 


7° 


S6° 


5 


5° 



200 



BROWN & SHARPE MFG. CO. 
NATURAL TANGENTS AND COTANGENTS 



/ 


35° 


36° 


37° 


38° 


39° 


/ 


Tang: 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 





.70021 


1.42815 


•72654 


1.37638 


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1.32704 


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1.27994 


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1.23490 


60 


I 


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1.42726 


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1.37554 


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1.32624 


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1.27917 


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1.23416 


59 


2 


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1.42638 


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1.37470 


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1.32544 


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1.27841 


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1-23343 


S8 


3 


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1.42550 


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1.37386 


• 75492 


1.32464 


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1.27764 


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1.23270 


57 


4 


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1.42462 


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1.37302 


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S6 




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1.32304 


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1.27611 


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1.23123 


55 


6 


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1. 37134 


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1.32224 


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1.27535 


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1.23050 


54 


7 


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I.37CSO 


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53 


8 


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1.22904 


52 


9 


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1.42022 


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1.36883 


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1.27306 


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SI 


10 


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SO 


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1.36716 


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49 


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48 


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47 


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45 


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44 


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43 


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41 


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37 


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36 


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1^35554 


.76502 


1.30716 


.79306 


1.26093 


.82190 


1.21670 


35 


26 


.71154 


1.40540 


• 73816 


I.3S472 


.76548 


1.30637 


.79354 


1.26018 


.82238 


1.21598 


34 


27 


.71198 


1.40454 


.73861 


1.35389 


.76594 


1.30558 


.79401 


1.25943 


.82287 


1.21526 


33 


28 


.71242 


1.40367 


.73906 


1.35307 


.76640 


1.30480 


.79449 


1.25867 


.82336 


1. 21454 


32 


29 


.71285 


1. 40281 


• 73951 


1.35224 


.76686 


1.30401 


.79496 


1.25792 


.82385 


1. 21382 


31 


30 


.71329 


1.40195 


.73996 


1.35142 


76733 


1.30323 


.79544 


1.25717 


.82434 


1.21310 


30 


31 


.71373 


1.40109 


.74041 


1.35060 


.76779 


I 30244 


.79591 


1.25642 


.82483 


1.21238 


29 


32 


.71417 


1.40022 


.74086 


1.34978 


.76825 


1.30166 


.79639 


1.25567 


.82531 


1.21166 


28 


33 


.71461 


1.39936 


.74131 


1.34896 


.76871 


1.30087 


.79686 


1.25492 


.82580 


1.21094 


27 


34 


.71505 


1-39850 


• 74176 


1.34814 


.76918 


1.30009 


.79734 


1. 25417 


.82629 


I. 21023 


26 


35 


.71549 


1.39764 


.74221 


1.34732 


.76964 


1.29931 


.79781 


1.25343 


.82678 


1.20951 


25 


36 


.71593 


1.39679 


.74267 


1.34650 


.77010 


1.29853 


.79829 


1.25268 


.82727 


1.20879 


24 


37 


.71637 


1.39593 


.74312 


1.34568 


.77057 


1.29775 


.79877 


1.25193 


.82776 


1.20808 


23 


38 


.71681 


1.39507 


.74357 


1.34487 


.77103 


1.29696 


.79924 


1. 25118 


.82825 


1.20736 


22 


39 


.71725 


1.39421 


.74402 


1.34405 


.77149 


1.29618 


.79972 


I -25044 


.82874 


1.20665 


21 


40 


.71769 


1.39336 


.74447 


1.34323 


.77196 


I. 29541 


.80020 


1.24969 


.82923 


1.20593 


20 


41 


.71813 


1.39250 


.74492 


1.34242 


.77242 


1.29463 


.80067 


1-24895 


.82972 


1.20522 


19 


42 


.71857 


1.39165 


.74538 


1.34160 


.77289 


1.29385 


.80115 


1.24820 


.83022 


1.20451 


18 


43 


.71901 


1.39079 


.74583 


1.34079 


.77335 


1-29307 


.80163 


1.24746 


.83071 


1-20379 


17 


44 


.71946 


1.38994 


.74628 


1.33998 


.77382 


1.29229 


.80211 


1.24672 


.83120 


1.20308 


16 


45 


.71990 


1.38909 


.74674 


1. 33916 


.77428 


1-29152 


.80258 


1-24597 


.83169 


1.20237 


15 


46 


.72034 


1.38824 


• 74719 


1.33835 


.77475 


1.29074 


.80306 


1-24523 


.83218 


1.20166 


14 


47 


.72078 


1.38738 


.74764 


1^33754 


.77521 


1.28997 


.80354 


1-24449 


.83268 


1.20095 


13 


48 


.72122 


1.38653 


.74810 


1^33673 


.77568 


1.28919 


.80402 


1.24375 


.83317 


1.20024 


12 


49 


.72167 


1.38568 


.74855 


1^33592 


.77615 


1.28842 


.80450 


1.24301 


-83366 


1.19953 


II 


50 


.72211 


1.38484 


.74900 


1.33511 


.77661 


1.28764 


.80498 


1.24227 


.83415 


1.19882 


10 


51 


.72255 


1.38399 


.74946 


1-33430 


.77708 


1.28687 


.80546 


1.24153 


.8346s 


1.19811 


9 


52 


•72299 


1. 383 1 4 


.74991 


1-33349 


.77754 


1,28610 


.80594 


1.24079 


.83514 


1.19740 


8 


S3 


■^''Hi 


1.38229 


• 75037 


l^33268 


.77801 


1-28533 


.80642 


1.24005 


.83564 


1. 1 9669 


7 


54 


.72388 


1.38145 


.75082 


I-33187 


.77848 


1.28456 


.80690 


1-23931 


.83613 


1.19599 


6 


SS 


.72432 


1.38060 


• 75128 


1-33107 


.77895 


1.28379 


.80738 


1-23858 


.83662 


1.19528 


5 


S6 


■72477 


1.37976 


•75173 


1 -33026 


.77941 


1.28302 


.80786 


1-23784 


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1.19457 


4 


H 


.72521 


1. 37891 


•75219 


1.32946 


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1.28225 


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1.23710 


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1. 19387 


3 


S8 


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1^37807 


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1-32865 


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I. 28148 


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1.23637 


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1.19316 


2 


f^ 


.72610 


1.37722 


.75310 


1-32785 


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1.28071 


.80930 


1.23563 


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1. 19246 


I 


6o 


•72654 


1^37638 


.75355 


1-32704 


.78129 


1.27994 


.80978 


1.23490 


.83910 


1.19175 





/ 


Cotang 


Tangr 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


5^ 


1° 


52 


° 


52 


,0 


51 





5C 


)° 



201 



BROWN & SHARPE MFG. CO. 

NATURAL TANGENTS AND COTANGENTS 



/ 


40° 


41° 


42- 


43° 1 


44° 


/ 




Tang- 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang ' 


rang 


Cotang 


o 


83910 


1.19175 


.86929 


1.15037 


.90040 


1.11061 


.93252 


1.07237 


96569 


1.03553 


60 


I 


83960 


1. 1910s 


.86980 


I. 14969 


.90093 


1. 10996 


.93306 


1.07174 


96625 


1.03493 


59 


2 


84009 


1.1903s 


.87031 


1.14902 


.90146 


1. 10931 


.93360 


1. 0711.2 


96681 


1.03433 


58 


3 


84059 


1.18964 


.87082 


1.14834 


.90199 


1.10867 


.93415 


1.07049 


96738 


1.03372 


57 


4 


84108 


I. I 8804 


.87133 


1.14767 


.90251 


1.10802 


.93469 


1.06987 


96794 


1.03312 


56 


5 


84158 


I. 18824 


.87184 


I. 14699 


.90304 


1.10737 


.93524 


1.06925 


96850 


1.03252 


55 


6 


84208 


1.18754 


.87236 


1.14632 


•90357 


1. 10672 


.93578 


1.06862 


96907 


1. 03192 


54 


7 


84258 


1. 1 8684 


.87287 


1.14565 


.90410 


1. 10607 


.93633 


1.06800 


96963 


1.03132 


53 


8 


84307 


1.18614 


.87338 


1. 14498 


.90463 


1. 10543 


.93688 


1.06738 


97020 


1.03072 


52 


9 


84357 


1. 18544 


.87389 


1.14430 


.90516 


I. 10478 


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1.06676 


97076 


1. 03012 


SI 


lO 


84407 


1. 18474 


.87441 


1.14363 


.90569 


1.10414 


•93797 


1. 066 1 3 


97133 


1.02952 


50 


II 


84457 


1. 1 8404 


.87492 


1. 14296 


.90621 


1.10349 


.93852 


1.06551 


97189 


1.02892 


49 


12 


84507 


1. 18334 


.87543 


1. 14229 


.90674 


1.10285 


.93906 


1.06489 


97246 


1.02832 


48 


13 


84556 


1. 18264 


.87595 


1. 14162 


.90727 


1.10220 


.93961 


1.06427 


97302 


1.02772 


47 


14 


84606 


1.18194 


.87646 


1.1409s 


.90781 


1.10156 


.94016 


1.06365 


97359 


1.02713 


46 


IS 


84656 


1.18125 


.87698 


1. 14028 


.90834 


i.ioogi 


.94071 


1.06303 


97416 


1.02653 


45 


i6 


84706 


1. 18055 


.87749 


1.13961 


.90887 


1. 10027 


.94125 


1.06241 


97472 


1.02593 


44 


17 


84756 


i.i79«6 


.87801 


1.13894 


.90940 


1.09963 


.94180 


1. 06179 


97529 


1.02533 


43 


i8 


84806 


1.17916 


.87852 


1. 13828 


.90993 


1.09899 


.94235 


1.06117 


97586 


1.02474 


42 


19 


84856 


1. 1 7846 


.87904 


1.13761 


.91046 


1.09834 


.94290 


1.06056 


97643 


1.02414 


41 


20 


84906 


1. 17777 


.87955 


1.13694 


.91099 


1.09770 


.94345 


1. 05994 


97700 


I.023SS 


40 


2X 


84956 


I. 17708 


.88007 


1. 13627 


.91153 


1.09706 


.94400 


1.05932 


97756 


1.02295 


39 


22 


85006 


1. 1 7638 


.88059 


1. 13561 


.91206 


1.09642 


.94455 


1.05870 


97813 


1.02236 


38 


23 


85057 


1. 1 7569 


.88110 


1.13494 


.91259 


1.09578 


.94510 


1.05809 


97870 


1.02176 


37 


24 


85107 


1.17500 


.88162 


1. 13428 


.91313 


1.09514 


.94565 


1.05747 


97927 


1.02117 


36 


25 


85157 


1. 1 7430 


.88214 


1.13361 


.91366 


1.09450 


.94620 


1.05685 


97984 


1.02057 


35 


26 


85207 


1.17361 


.88265 


1.1329s 


.91419 


1.09386 


.94676 


1.05624 


98041 


1.01998 


34 


27 


85257 


1. 17292 


.88317 


1.13228 


.91473 


1.09322 


•94731 


1.05562 


98098 


1.01939 


33 


28 


85308 


1. 17223 


.88369 


I. 13162 


.91526 


1.09258 


.94786 


1.05501 


98155 


1.01879 


32 


29 


85358 


1.17154 


.88421 


1.13096 


.91580 


1.09195 


.94841 


1. 05439 


98213 


1.01820 


31 


30 


85408 


1.1708s 


.B8473 


1. 13029 


.91633 


1.09131 


.94896 


1.05378 


98270 


1.01761 


30 


31 


85458 


1.17016 


.88524 


I. I 2963 


.91687 


1.09067 


.94952 


I.05317 


98327 


1.01702 


29 


32 


85509 


1.16947 


.88576 


1. 1 2897 


.91740 


1.09003 


.95007 


1.05255 


98384 


1.01642 


28 


33 


85559 


1.16878 


.88628 


1.12831 


.91794 


1.08940 


.95062 


1.05194 


98441 


1.01583 


27 


34 


85609 


1.16809 


.88680 


1.12765 


.91847 


1.08876 


.95118 


I.OS133 


98499 


1.01524 


26 


35 


85660 


1.16741 


.88732 


1.12699 


.91901 


1.08813 


.95173 


1.05072 


98556 


1. 01465 


25 


36 


85710 


1.16672 


.88784 


1. 12633 


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1.08749 


.95229 


1.05010 


98613 


1. 01406 


24 


37 


85761 


1.16603 


.88836 


1. 12567 


.92008 


1.08686 


.95284 


1.04949 


98671 


1.01347 


23 


38 


8581 1 


1-16535 


.88888 


1.12501 


.92062 


1.08622 


.95340 


1.04888 


98728 


1.01288 


22 


39 


85862 


1. 16466 


.88940 


1.12435 


.92116 


1. 08559 


•95395 


1.04827 


98786 


1.01229 


21 


40 


85912 


1. 16398 


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1.12369 


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1.08496 


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1.04766 


98843 


1.01170 


20 


41 


85963 


I. 16329 


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1.12303 


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1.08432 


.95506 


1. 0470s 


98901 


1.01112 


19 


42 


86014 


1.16261 


.89097 


1. 12238 


.92277 


1.08369 


.95562 


1.04644 


98958 


1. 01053 


18 


43 


86064 


1. 16192 


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1.12172 


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1.08306 


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1.04583 


99016 


1.00994 


17 


44 


86115 


1. 16124 


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1.12106 


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1.08243 


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1.04522 


99073 


1.00935 


16 


45 


86166 


1.16056 


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1.12041 


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1.08179 


.95729 


1. 04461 


99131 


1.00876 


15 


46 


86216 


1. 15987 


.89306 


1.11975 


.92493 


1.08116 


.95785 


1. 04401 


99189 


1.00818 


14 


47 


86267 


1.15919 


.89358 


1.11909 


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1.08053 


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1.04340 


99247 


1.00759 


13 


48 


86318 


1.15851 


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1.11844 


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1.07990 


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1.04279 


99304 


1. 00701 


12 


49 


86368 


I. 15783 


.89463 


1.11778 


.92655 


1.07927 


.95952 


1. 04218 


99362 


1.00642 


II 


50 


86419 


1.15715 


.89515 


1.11713 


.92709 


1.07864 


.96008 


1. 04158 


99420 


1.00583 


10 


51 


86470 


1.15647 


.89567 


1.11648 


.92763 


1. 07801 


.96064 


1.04097 


99478 


1.0052s 


9 


52 


86521 


1.ISS79 


.89620 


1.11582 


.92817 


1.07738 


.96120 


1.04036 


99536 


1.00467 


8 


53 


86572 


1.15511 


.89672 


1.11517 


.92872 


1.07676 


.96176 


1.03976 


99594 


1.00408 


7 


54 


86623 


1.15443 


.89725 


1.11452 


.92926 


1.07613 


.96232 


1.03915 


99652 


i.oqsso 


6 


55 


86674 


1.1S37S 


.89777 


1. 11387 


.92980 


1.07550 


.96288 


1.03855 


99710 


1. 00291 


S 


56 


86725 


1.15308 


.89830 


1.11321 


.93034 


1.07487 


.96344 


1.03794 


99768 


1.00233 


4 


57 


86776 


I. 15240 


.89883 


1.11256 


.93088 


1.0742s 


.96400 


1.03734 


99826 


I. 001 75 


3 


58 


86827 


1.15172 


.89935 


1. 11191 


.93143 


1.07362 


.96457 


1.03674 


99884 


1.00116 


2 


59 


86878 


1.IS104 


.89988 


1. 11126 


.93197 


1.07299 


.96513 


1.03613 


99942 


1.00058 


I 


60 


86929 


1.15037 


.90040 


1.11061 


.93252 


1.07237 


.96569 


1.03553 I 


00000 


1.00000 





C 


otang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang C 


otang 


Tang 


/ 


/ "~ 


4c 


)° 


4^ 


r 


4; 


7° 


4^ 


5° 


45° 



202 



BROWN & SHARPE MFG. CO. 



DECIMAL EQUIVALENTS OF PARTS OF AN INCH 



J^ ... .01563 

A 03125 

A ... .04688 
i-i6 0625 

^5^ ... .07813 

A- 09375 

7- ... .10938 
1-8 125 

^\ ... .14063 

-352 15625 

H ... .17188 
3-16 1875 

if ... .20313 

3V 21875 

if ... .23438 
1-4 25 

i| ... .26563 

A 28125 

if ... .29688 
5-16 3125 



IJ ... .32813 

4i 34375 

If ... .35938 
3-8 375 

II ... .39063 

i| 40625 

IJ ... .42188 
7-16 4375 

If ... .45313 

If 46875 

|i ... .48438 
1-2 5 

ff ... .51563 
H 53125 

M ... .54688 

6 4 

9-16 5625 

ff ... .57813 

if 59375 

If ... .60938 
5-8 .625 

If ... .64063 

21- ...... .65625 

43 ... .67188 

11-16 6875 



45 

6? 



64 



3-4 



49 

64 



,2 5 
32 



64 



13-16 



53 
64 



2.7. 
32 



55 
6¥ 



7-8 



Al 
64 



29 
32" 



59. 
64 



15-16 



31 
'3 2- 



13 
64 



. .70313 

. .71875 
. .73438 

. 75 



.76563 

.78125 
.79688 
.8125 

.82813 
.84375 
.85938 

.875 



. .89063 
. .90625 
. .92188 
. .9375 



, .95313 
.96875 
.98438 

1.00000 



203 



BROWN & SHARPE MFG. CO. 



TABLE OF DECIMAL EQUIVALENTS 

OF 

MILLIMETRES AND FRACTIONS OF MILLIMETRES. 



mm. Inches. 


mm. Inches. 


mm. Inches. 


mm. Inches. 


, ^ = .00039 


Wo = -01399 


Wo = -03530 


m = -03740 


i§0 = .00079 


fj = .01339 


^ = .03559 


fo = .03780 


il = -00118 


Wo = -01378 


Wo = -02598 


fo = .03819 


m = -00157 


1) = -01417 


^ = .03638 


fo = .03858 


4 = .00197 


#0 = -01457 


m = -03677 


^ = .03898 


4 = .00336 


m = -01496 


i = .03717 


1 = .03937 


^ = .00376 


^ = .01535 


fo = .03756 


3 = .07874 


4 = .00315 


Wo = .01575 


^ = .03795 


3 = .11811 


4 = .00354 


^0 = -01614 


fo = .03835 


4 =- .15748 


Wo = -00394 


fo = -01654 


fo = .02874 


5 = .19685 


^ = .00433 


f == .01693 


fo = .02913 


6 = .33633 


i) = .00473 


i) = .01733 


ifo = .03953 


7 = .37559 


m = .00513 


Wo = -01773 


f = .03993 


8 = .31496 


^ = .00551 


f, = .01811 


fo = .03033 


9 = .35433 


ij = .00591 


f = .01850 


f = -03071 


10 = .39370 


m = -00630 


f = .01890 


f = .03110 


11 = .43307 


1^ = .00669 


^ = .01939 


i; == .03150 


13 = .47344 


^ = .00709 


j^ = .01969 


fo = .03189 


13 = .51181 


Wo = -00748 


iS) = .03008 


fo = .03338 


14 = .55118 


Wo = .00787 


Wo = .03047 


fo = .03368 


15 = .59055 


^ = .00837 


Wo = .03087 


fo = .03307 


16 = .63993 


il = -00866 


fo = .03136 


m = .03346 


17 = .66939 


H = .00906 


^ = .03165 


fo = .03386 


18 = .70866 


^ = .00945 


i^ = .03305 


fo = .03435 


19 = .74803 


^ = .00984 


fo = .03344 


fo = .03465 


30 = .78740 


Wo = .01034 


^ = .03383 


fj = .03504 


31 = .83677 


1^ = .01063 


fo = .03333 


fo = .03543 


33 = .86614 


fo = -01103 


^ = .03363 


f = .03583 


33 = .90551 


m = -01143 


^ = .03403 


fo = .03633 


34 = .94488 


fi - -01181 


Wo - -O^il 


S = .03661 


35 = .98435 


^ = .01330 


^ - .03480 


Wo = .03701 


36 =1.03363 


Wo = .01360 









10 mm. = 1 Centimeter : 
10 cm. = 1 Decimeter = 



0.3937 inches. 
J.937 inches. 



10 dm. = 1 Meter = 39.37 inches. 
25.4 mm. = 1 English Incli. 



204 



BROWN & SHARPE MFG. CO. 

OTHER PUBLICATIONS 

ISSUED BY THE BROWN & SHARPE MFG. CO. 



Practical Treatise on Milling and Milling Machines 

Edition of 1919 

This work is a thorough treatise on MilHng and MiUing 
Machines. 332 pages, 210 illustrations. Sent on receipt 
of price. Cloth, $1.50; Cardboard, $1.00. 



Construction and Use of Automatic Screw Machines 

Edition of 1919 

This book is published to assist those who are not 
familiar with the construction and use of the Automatic 
Screw Machine. Sent on receipt of price. Cardboard, 
50 cents. 

Construction and Use of Universal Grinding 
Machines 

Edition of 1919 

This work describes the construction and use of 
Universal Grinding Machines, as made by us. Fully 
illustrated. Sent on receipt of price. Cardboard, 25 cents. 



Construction and Use of Plain Grinding Machines 

Edition of 1920 

This work describes the construction and use of Plain 
Grinding Machines, as made by us. Fully illustrated. 
Sent on receipt of price. Cardboard, 25 cents. 



Formulas in Gearing 

Edition of 1918 

This work supplements the * 'Practical Treatise on 
Gearing" and contains formulas for solving the problems 
that occur in gearing. Sent on receipt of price. Cloth, 
$1.50. 

Hand Book for Apprenticed Machinists 

This book, illustrated, is for the Apprenticed Machinist. 
It is carefully written to assist the learner in the use of 
Machine Tools. Sent on receipt of price. Cloth, 75 cents. 

205 



BROWN & SHARPE MFG. CO. 

INDEX 
A 

PAGE 

Abbreviations of Parts of Teeth and Gears 12 

Addendum 11 

Angle, How to Lay Off an 146, 163 

Angle Increment 161 

Angle of Edge 157 

Angle of Face 159 

Angle of Pressure 164 

Angle of Spiral 114 

Angular Velocity 10 

Annular Gears 43, 45 

Arc of Action 165 

Automatic Gear Cutting Machine 156 

B 

Base Circle 17 

Base of Epicycloidal System 36 

Base of Internal Gears 45 

Bevel Gear Blanks 48 

Bevel Gear Cutting on Automatic Machine 73 

Bevel Gear Angles by Diagram 50 

Bevel Gear Angles by Calculation 157 

Bevel Gear, Form of Teeth of 55 

Bevel Gear, Spiral 79 

Bevel Gear, Whole Diameter of 50, 161 

C 

Centres, Line of 11 

Chordal Thickness 174, 175 

Circular Pitch, Linear or 12 

Circular Pitch, "Nuttall" 177 

Classification of Gearing 13 

Clearance at Bottom of Space 14 

Clearance in Pattern Gears 16 

207 



BROWN & SHARPE MFG. CO. 
INDEX 

PAGE 

Condition of Constant Velocity Ratio 10 

Contact, Arc of 165 

Continued Fractions 166 

Coppering Solution 107 

Cosine, Sine and 151 

Cutters, How to Order 105 

Cutters, Table of Epicycloidal 105 

Cutters, Table of Involute 104 

Cutters, Table of Feeds for 103 

Cutters, Shape of 121 

Cutting Bevel Gears on Automatic Machine 73 

Cutting Spiral Gears in a Universal Milling Machine. 124 

D 

Decimal Equivalents, Tables of 203, 204 

Diameter Increment 159 

Diameter of Pitch Circle 14 

Diameter Pitch 14 

Diametral Pitch 25 

Diametral Pitch, "Nuttall" 176 

Distance between Centres 15 

E 

Elements of Gear Teeth 13 

Epicycloidal Gears, with more or less than 15 Teeth 41 

Epicycloidal Gears, with 15 Teeth 36 

Epicycloidal Rack 38 

F 

Face, Width of Spur Gear 102 

Flanks of Teeth in Low Numbered Pinions 29 

G 

Gear Cutters, How to Order 105 

Gear Patterns 16 

Gearing Classified 13 

Gears, Bevel. : 48, 55, 157 

Gears, Epicycloidal 36 

208 



BROWN & SHARPE MFG. CO. 
INDEX 

PAGE 

Gears, Involute 17 

Gears, Spiral 108, 114, 117, 124 

Gears, Spiral Bevel 79 

Gears, Strength of 135 

Gears, Worm 83 

H 

Herring-bone Gears 132 

Hobs 87 

I 

Increment, Angle 161 

Increment, Diameter 159 

Interchangeable Gears 34 

Internal or Annular Gears 43, 45 

Involute Gears, 30 Teeth and over 17 

Involute Gears, with Less than 30 Teeth 29 

Involute Rack 20 

L 

Land 10 

Lead of a Worm 83 

Length of a Worm and a Hob 98 

Limiting Numbers of Tee thin Internal Gears 43 

Line of Centres 11 

Line of Pressure 20, 164 

Linear or Circular Pitch 12 

Linear Velocity 9 

M 

Machine for Cutting Bevel Gears 69 

Method for Obtaining Set-over for Bevel Gears 70 

Method of Drawing a Rack 21 

Method of Laying Out Single-curve Gears 17 

Mitre Gears 78 

Module 14 

209 



BROWN & SHARPE MFG. CO. 

INDEX 

N 

PAGE 

Normal 117 

Normal Helix 117 

Normal Pitch 119 

O 

Original Cylinders 9 

P 

Pattern Gears 16 

Pinions '. 43 

Pitch Circle 12 

Pitch, Circular or Linear 12 

Pitch, a Diameter 14 

Pitch, Diametral 25 

Pitch, Normal 117 

Pitch of Spirals 112 

Polygons, Calculations for Diameters of 153 

R 

Rack 20 

Rack for Epicycloidal Gears 38 

Rack for Involute Gears 21- 

Rack for Spiral Gears .;.... 119 

Relative Angular Velocity , .10 

RolHng Contact of Pitch Circle 12 

S 

Screw Gearing 108 

Shape of Cutters 121 

Sine and Cosine 151 

Single-Curve Teeth 17 

Speed of Gear Cutters 102 

Spiral Bevel Gears 79 

Spiral Gearing 108, 114, 117, 124 

Standard Templets 38 

Standard Proportions for Spur Gears 143 

210 



BROWN & SHARPE MFG. CO. 
INDEX 

PAGE 

Strength of Gears 135 

Squares and Square Roots 172 

T 

Table of Chordal Thickness of Gear Teeth 174, 175 

Table of Decimal Equivalents 203, 204 

Table of Feeds for Gear Cutters 103 

Table of Gear Strength Factors 134 

Table for Obtaining Set-over for Bevel Gears. . 70 

Table for Right Angled Triangles 182 

Table of Sines, etc 184-202 

Table of Tooth Parts 178-181 

Tangent of Arc and Angle 145 

V 

Velocity, Angular 10 

Velocity, Linear 9 

Velocity, Relative 10 

W 

Wear of Teeth 102, 131 

Worm Gears 83 

Worm, Length of 98 



211 



